MECÁNICA DE ROCAS APLICADA A LA MINERÍA
MODELO DE FALLA CIRCULAR
ÍNDICE 1.
INTRODUCCI INTRODUCCIÓN.... ÓN............ ............... .............. ............... ............... ............... ............... ............... ............... .............. ............... ............... ............... ............... ....................... ....................2 ....2
2.
RESUMEN...... RESUMEN............. .............. ............... ............... ............... ............... .............. ............... ............... ............... ............... ............... ............... .............. ............................ ..............................3 .........3
3.
OBJETIVOS... OBJETIVOS.......... ............... ............... .............. ............... ............... ............... ............... ............... ............... .............. ............... ............... ............... ............... .............................. .......................44
4.
MARCO TEÓRICO.... TEÓRICO............ ............... .............. ............... ............... ............... ............... .............. ............... ............... ............... ............... .............. ............... ............... ............... ...........5 ...5 4.1
DEFINICIÓN DEFINICIÓN DE FALLA FALLA CIRCULAR... CIRCULAR.......... .............. ............... ............... ............... ............... .............. ............... ............... .................. .................5 ......5
4.2
TIPO TIPO DE MAT MATERIAL ERIAL Y MORFO MORFOLOG LOGIA IA EN FALLA FALLA CIRCU CIRCULAR LAR.... ........ ....... ....... ........ ....... ....... ......... .......... .......... .....66
4.3
SUPERF SUPERFICI ICIES ES CIRCUL CIRCULARE ARES S DE DESLIZAMI DESLIZAMIENT ENTO.. O...... ........ ....... ....... ........ ........ ....... ....... ........ ......... .......... .......... .......... ........ ...
4.4
VELOCIDAD VELOCIDAD EN UNA FALLA FALLA CIRCULAR.. CIRCULAR.......... ............... .............. ............... ............... ............... ............... .............. ...................... ...............!!
4.5
PART PARTES ES DE UNA FALLA FALLA CIRCULAR.. CIRCULAR......... .............. ............... ............... ............... ............... .............. ............... ............... ........................" ................."
5.
FACTOR FACTORES ES CONDICIO CONDICIONANTES NANTES Y DESENCADENAN DESENCADENANTES TES DE UNA UNA FALLA FALLA CIRCULAR... CIRCULAR.....11 ..11
6.
RESISTENCIA RESISTENCIA AL CORTE CORTE EN MAT MATERIALES ERIALES ALT ALTAMENTE AMENTE FRACTURA FRACTURADOS...... DOS.......................12 .................12
.
FACTOR FACTOR DE SEGURIDAD SEGURIDAD........ ............... .............. ............... ............... ............... ............... .............. ............... ............... ............... ................................. ............................13 ...13
!.
.1
FACTOR ACTOR DE SEGU SEGURID RIDAD AD DEFI DEFINID NIDO O COMO COMO RELACI RELACIÓN ÓN DE DE MOMEN MOMENTOS TOS.... ......... .......... ........14 ...14
.2
FACTOR ACTOR DE SEGU SEGURID RIDAD AD DEFI DEFINID NIDO O COMO COMO UNA REDUCC REDUCCIÓN IÓN DE LA LA RESITEN RESITENCIA CIA AL CORTE LÍMITE DEL TERRENO.........................................................................................16
M#TODOS M#TODOS DE C$LCULO...... C$LCULO............. .............. ............... ............... ............... ............... .............. ............... ............... ............... ............... .............. ................ .................1 ........1 DEFINICIÓN DE LOS M#TODOS DE C$LCULO............................................................................1
".
M#TODO M#TODO DE DE E%UILIBR E%UILIBRIO IO LÍMITE LÍMITE....... ............... ............... ............... ............... .............. ............... ............... ............... .......................... ...........................2& .........2& ".1
E%UILIBRIO E%UILIBRIO GLOBAL DE LA MASA......... MASA................ ............... ............... ............... ............... .............. ............... ............... ............... ...............22 .......22
".1.1
ANALISIS NO NO DR DRENADO '( '()C*+ ,)&-.........................................................................23
".1.2 ".1.2
ANALIS ANA LISIS IS GENERA GENERAL L 'DRENA 'DRENADODO- CIRCUL CIRCULO O DE ROZAMI ROZAMIENT ENTO... O........ .......... .......... ...........2 ......233
".1. ".1.33
SIST SISTEMA EMAT TIZAC IZACIO ION N DEL DEL METO METODO DO DE CIR CIRCU CULO LO DE DE ROZ ROZAM AMIE IENT NTO O SOLUCIONES MEDIANTE ABACOS...........................................................................2!
".2
M#TODO M#TODO DE DOV DOVELAS ELAS O REBANA REBANADAS DAS.... ....... ....... ........ ....... ....... ........ ........ ....... ....... ........ ....... ....... ........ ....... ......... ........... .......... .........4 ....4
".2.1 ".2.1
M#TODO M#TODOS S APRO/ APRO/IMA IMADOS DOS.... ....... ....... ........ ....... ....... ........ ........ ....... ....... ........ ....... ....... ........ ....... ....... ........ ........ ....... ....... ........ ......... .......... .......5 ..533
M#TODO DE FELLENIUS O 0CONVENCIONAL......................................................................53 M#TODO DE BISOP SIMPLIFICADO.........................................................................................55 M#TODO SIMPLIFICADO DE JANBU..........................................................................................5" ".2.2 ".2.2
M#TODO M#TODOS S PRECIS PRECISOS OS O COMPLE COMPLETOS TOS... ....... ....... ....... ........ ........ ....... ....... ........ ....... ....... ........ ....... ....... ........ ........ ....... ....... ......... ......6 .6!!
M#TODO DE MORGENSTERN PRIC '1"65-............................................................................6" M#TODO DE SPENCER '1"6-.......................................................................................................3 1&
CONCLUSIO CONCLUSIONES.... NES............ ............... ............... ............... .............. ............... ............... ............... ............... .............. ............... ............... ............... ............... ..................... ..............3 3
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MODELO DE FALLA CIRCULAR
11
BIBLIOGRAF BIBLIOGRAFIA..... IA............ .............. ............... ............... ............... ............... .............. ............... ............... ............... ............... ............... ............... .............. ....................... ................44
1. INT INTRODUC RODUCCIÓN CIÓN C 789 9: 8*9; (<=>7?9;> @9: >7:*@ =9;>9 7 * <?8:9 7::7 '=>*=7=9<9=(*>9 =9<9=(*>9 9;=99;>9 9; : (> 9; :7 9*=?@7@+ 9*=?@7@+ 9 <= 9:: *9 :7 ?;9>?7(?;9 ?;9>?7(?;9 @9 (7< '?;?>*- @9 :78=7>=? @989; 9= *?(?9;>9 ( <7=7 <@9= (7=7(>9=?7= 9; :7 9@?@7 @9 : <?8:9 :7 (7=7(>9=>?(7 99(H;?(7 @9: 7(? =(+ 7 ( : <?8:9 9(7;? @9 =>*=7. D98?@ 7 :7 ?;9>78?:?@7@ @9 >7:*@9+ 9 <=@*(9; 7=? >?< @9 7::7 '=>*=7-+ 9>7 (*==9; 7 97 <= @9:?7?9;>+ @9<=9;@??9;> *9:(. D9;>= @9 : >?< @9 7::7 <@9 9;(?;7= 7 :7 7::7 <:7;7=+ 7::7 @9 *9:(+ 7::7 (?=(*:7=. E: <=99;>9 >=787 >=7>7=7 9(:*?79;>9 @9: >?< @9 7::7 (?=(*:7=. P= 7::7 (?=(*:7= 9 9;>?9;@9 9;>?9;@9 *9 9 :7 *<9=?(?9 @9 @9:?7?9;> @9:?7?9;> 79978:9 79978:9 7 *;7 *<9=?(?9 (?:;@=?(7 (?:;@=?(7 (*7 >=7;9=7: >=7;9=7: 9 7997 7 *; 7=( @9 (=(*:. E> 9 @?(9 7 *9 9 <*9@9 @9>=7= *9 9; *9: K;9 :7 *<9=?(?9 @9 =>*=7 9 *;7 9?(7 *9 <= : >7;> 9 7<=?7 *(K 7 *; (=(*:. L7 7=7 @9 7;H:?? *9:9; <7=>?= @9 :7 K?<>9? @9 *9 :7 *<9=?(?9 @9 =>*=7 @9:?7?9;> 9 (?=(*:7= 9>7 7 * 9 <7=>9; @9 *;7 (7=7(>9=?7(?; 99(H;?(7 @9: 7(? =(. E: 7;H:?? @9 7::7 (?=(*:7= 9 =97:?7 7 7*9:: <9=?:9 @9 >7:*@ 9; ;7 9; @;@9 9: 7(? =( <=99;>9 *; GSI 9;= 7 3&. P7=7 =97:?7= 9> 7;H:?? 9 (;?@9=7; (;@?(?;9 9>H>?(7 7 ( <9*@9>7>?(7 '@?;H?(7-. L >@ @9 (H:(*: <7=7 @9>9=?;7= 9: =7@ @9 9*=?@7@ <*9@9; 9= 7=?+ 7:*; H 97(> *9 >=. E; (;(:*?;+ 9 @989 (;(9= 9: =7@ @9 9*=?@7@ <7=7 @9>9=?;7= *;7 <?8:9 7::7 (?=(*:7= 9; 9: >7:*@+ <7=7 :*9 9= (;>=:7@.
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MODELO DE FALLA CIRCULAR
2. RE RESUM SUMEN EN E; *9: *9: 7(? 7(? =( =( @9 877 877 (7:?@ (7:?@7@ 7@ 7:>7 7:>79 9;> ;>99 =7(> =7(>*=7 *=7@ @ ?; ?; @?=9( @?=9((? (?;9 ;9 <=9@?;7;>9 @9 @?(;>?;*?@7@+ :7 =>*=7 9 <=@*(9 7 : :7= @9 *;7 *<9=?(?9 @9 @9:?7?9;> ?;>9=;7 ?*?9;@ :7 :;97 @9 9;= =9?>9;(?7. L (=(*: @9 =>*=7 *9:9; <77= <= 9: 7:*@+ 7*;*9 >78?; 9 =??;7; 9; >=7 <7=>9 @?9=9;>9 @9: >7:*@+ 9; :7 (7=7(>9=>?(7 =9?>9;>9 @9: 7>9=?7:+ 7:>*=7 9 ?;(:?;7(?; @9: >7:*@+ 9>(. E: ??9;> >?9;9 *;7 ;7>*=7:97 =>7(?;7:+ 7:=9@9@= @9 *; 99 @?<*9> <7=7:9:79;>9 7: >7:*@. L >@ @9 (H:(*: *7@ ; : @9 9*?:?8=? :?>9+ 9; : *9 <@9 (;?@9=7= 7 :7 77 <>9;(?7:9;>9 @9:?78:9 ( *;7 :7 '>@ @9 9>78?:?@7@ :87:- @??@?=:7 9; 77 9=>?(7:9 '>@ @9 :7 @9:7-. D9;>= @9 : >@ @9 9>78?:?@7@ :87:+ 9 9;(*9;>=7 9: (=(*: @9 =7?9;> *9 ; (H:(*: 9; : *9 >78?; 9 <*9@9 9<:97= 9: * @9 H87(. A * 9 9: >@ @9 :7 @9:7 9 @??@9 9; (H:(*: 7<=?7@ 'J7;8*+ F9::9;?*+ B?K< ?<:??(7@-+ (H:(*: <=9(? 'M=9;>9=; P=?(9+ S<9;(9=+ B?K< =?*=-. T@ 9> (H:(*: ; *7@ <7=7 @9>9=?;7= 9: 7(>= @9 9*=?@7@ @9: 7(? =( <7=7 7 @9(7=>7= *;7 <?8:9 7::7 (?=(*:7=.
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MODELO DE FALLA CIRCULAR
3. OBJETIVOS
C<=9;@9= (*7;@+ ( @;@9 (*==9 *;7 7::7 (?=(*:7= A;7:?7= :7 *9=7 *9 7(>7; 9 9>9 >?< @9 7::7. E>?7= 9: 7(>= @9 9*=?@7@ 9@?7;>9 @?9= >@. A;7:?7= (<7=7= : @?9= >@ @9 (H:(*:.
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MODELO DE FALLA CIRCULAR
4. MARCO TEÓRICO 4.1
DEFINICIÓN DE FALLA CIRCULAR
E *;7 7::7 ?; (;>=: 9>=*(>*=7: 7 @?9=9;(?7 @9 :7 7::7 <:7;7+ @9 (*Q7 :(7?9;>. L7 7::7 (?=(*:7=+ >78?; (;(?@7 ( @9:?7?9;> =>*=7 =>7(?;7:+ 9 <=@*(?@7 <= *(K 9> @9 7::7+ :7 (*7:9 >?9;9; @?>?;> =*8 8*7?9;> 'F?*=7 1-+ @97; :7 =(7 * =7(>*=7@7 ?; @?=9((?;9 <=99=9;(?7:9 @9 @9:?7?9;>+ <9=@?9;@ :7 (K9?;. E>9 >?< @9 =>*=7 7@9H @9 <=@*(?=9 9; >9==9; K;9+ (*<:9 (; :7 (;@?(?; @9 *9 9: >77Q @9 :7 <7=>(*:7 @9 *9: =(7 97 * <9*9Q7 9; (<7=7(?; (; 9: >77Q @9: >7:*@. L7 =>*=7 9 <=@*(9 7 : :7= @9 *;7 *<9=?(?9 @9 @9:?7?9;> ?;>9=;7 ?*?9;@ :7 :;97 @9 9;= =9?>9;(?7. P= : >7;> 9 9;>?9;@9 *9 9>7 7<=?7(?; @9 =>*=7 9 K7(9 (*7;@ ; 9 <?8:9 @9>9=?;7= :7 7?:?7 @9 @?(;>?;*?@7@9 *9 (;>=:7 :7 ?;9>78?:?@7@+ 9 @9(?= ?; ?>97 @9 @?(;>?;*?@7@ ?@9;>??(78:9. 'F?*=7 2-.
Figura 1. Macizo de roca distintos rumbos y buzamientos (muy fracturada)
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MODELO DE FALLA CIRCULAR
Figura 2. &royección estereo'rfica de un talud con falla circular. L7 *<9=?(?9 @9 @9:?7?9;> 9 79978:9 7 *;7 *<9=?(?9 (?:;@=?(7 (*7 >=7;9=7: 9 7997 7 *; 7=( @9 (?=(*: 'F?*=7 3-.
Figura 3. Sección transversal de una superficie de rotura circular. E: ??9;> >?9;9 *;7 ;7>*=7:97 =>7(?;7:+ 7:=9@9@= @9 *; 99 @?<*9> <7=7:9:79;>9 7: >7:*@. 'F?*=7 4-
Figura 4. Rotura típica con forma cilíndrica (VARNS! "#$%).
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MODELO DE FALLA CIRCULAR
4.2
TIPO DE MATERIAL Y MORFOLOGIA EN FALLA
CIRCULAR L7 ;7>*=7:97 ?;>=;9(7 @9: 7>9=?7: 7;>?9;9 *;7 9>=9(K7 =9:7(?; (; 9: >?< @9 ?;9>78?:?@7@ *9 <*9@9; <=@*(?=9+ (;@?(?;7;@ <*@?9;@ 9>?7=9 @9 7;>97; :7 *(9<>?8?:?@7@ @9 (7@7 7>9=?7:+ 7 *9 9 @97==::9 *; ??9;> @9>9=?;7@. E; 9: <=99;>9 >=787 K78:7=9 @9 =>*=7 (?=(*:7= 9; macizos rocosos 7:>79;>9 =7(>*=7@+ 9; : *9 9 @97==::7; *<9=?(?9 @9 (=>9+ ??:7=9 7 :7 <=@*(?@7 9; *9:. N @989 (;*;@?= 78 >=?; 7(? =( *9:+ ?9;@ 9: <=?9= *; (;*;> @9 8:*9 9<7=7@ <= @?(;>?;*?@7@9 9: 9*;@ *; (;*;> @9 <7=>(*:7 :?@7+ *9:>7 <( (99;>7@7. L @9:?7?9;> =>7(?;7:9 '=>*=7 (?=(*:7=- *9:9; >9;9= *;7 =:7 >7;>9 ? 9: @9:?7?9;> 9 @9 =7; 9>9;?; 9; 9: 9;>?@ :;?>*@?;7: @9: >7:*@+ <*9@9; 7997=9 7 *;7 superficie cilíndrica @9 99 <7=7:9: 7: >7:*@ 'F?*=7 4-.
Figura 5. Rotura típica con forma de cucara (VARNS! "#$%).
4.3
SUPERFICIES CIRCULARES DE DESLIZAMIENTO
L (=(*: @9 =>*=7 *9:9; <77= <= 9: 7:*@+ 7*;*9 >78?; 9 =??;7; 9; >=7 <7=>9 @'7 -?9=9;>9 @9: >7:*@+ 9; :7 (7=7(>9=>?(7 =9?>9;>9 @9: 7>9=?7:+ 7:>*=7 9 ?;(:?;7(?; @9: >7:*@+ 9>(. 'F?*=7 6-
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MODELO DE FALLA CIRCULAR
Figura 6. *iferentes superficies circulares de deslizamiento (+im,nez Salas! +.A - Mollina! R.) '7- Circulo superficial de pie.- P77 <= 9: 7:*@ * <*;> H 87 9 9;(*9;>=7 9; @?(K 9 (7. E; >7:*@9 =7@ <= >9==9; (; ф 9@? 7 7:>. E; >7:*@9 @9 ф 9@? 7 87+ 7; ;*:+ ?9<=9 *9 * <9;@?9;>9 97 • •
?<=>7;>9+ 7= *9 *; 7:= *9 9 ?;@?(7 H 7@9:7;>9. '8- Circulo profundo.- P77 <= @987 @9: 7:*@. S9 <=@*(9 9; >7:*@9 >9;@?@ (; ф * 87 ;*:. •
'(- Circulo profundo de pie.- P77 <= 9: 7:*@ <9= <=*;@?7 <= @987 @9 9>9 9; 7:; <*;>. S9 <=@*(9 9; (7 ?;>9=9@? 9;>=9 '7- '8-. •
'@- Circulo de talud.- L7 :;97 @9 @9:?7?9;> 7:=7 9; :7 (7=7 @9: >7:*@.
4.4
VELOCIDAD EN UNA FALLA CIRCULAR
L7 9:(?@7@ @9 9>9 >?< @9 =>*=7 7=7 @9 :9;>7 7 @9=7@7+ >9;?9;@ =7; ?;:*9;(?7 :7 ?;(:?;7(?; @9 :7 *<9=?(?9 @9 =>*=7 9; 9: . S? 9: <9=?: @9 :7 *<9=?(?9 @9:?7@7 9 ?;(:?;7 K7(?7 9: ;>9+ 9 9=7 9: 9*?:?8=? 9; :7 77 ?;9>78:9+ @9(=9(?9;@ 9: 9;> ?;@*(>= <*@?;@9 ::97= 7 @9>9;9= 9: @9:?7?9;>.
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MODELO DE FALLA CIRCULAR
Figura 7. lasificación de los movimientos se'/n su velocidad.
Figura 8. 0ipos de movimientos y su ran'o de velocidad.
4.5
PARTES DE UNA FALLA CIRCULAR
U;7 7::7 (?=(*:7= >78?; (;(?@7 ( @9:?7?9;> =>*=7 =>7(?;7:+ (;>7 @9 7=?7 <7=>9 *9 9;(?;7=9 7 (;>?;*7(?; (; * ;9;(:7>*=7 7(>*7:. •
Escarpe principal.- S*<9=?(?9 *9 9 =7 8=9 9: >9==9; ; @9:?7@ 9; :7 <9=?9=?7 @9: @9:?7?9;>. S9 @97==::7 7 (7*7 @9: ??9;> @9: 7>9=?7:
•
@9:?7@ K7(?7 787 7@9;>= =9<9(> @9: 7>9=?7: ?;>7(>. Escarpe secundario.- S*<9=?(?9 9(7=<7@7 *9 9 =7 @9;>= @9: 7>9=?7: @9<:77@ @98?@ 7 ??9;> @?9=9;(?7:9 @9: ?. EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
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MODELO DE FALLA CIRCULAR
•
Cabeza.- P7=>9 *<9=?= @9: 7>9=?7: @9:?7@. Coronación.- L*7= 9>=?( @9 : <*;> H 7:> @9: (;>7(> 9;>=9 9:
•
7>9=?7: @9<:77@ 9: 9(7=<9 <=?;(?<7:. Pie de la superficie de rotura.- I;>9=9((?; @9 :7 <7=>9 ?;9=?= @9 :7 *<9=?(?9 @9
•
=>*=7 :7 *<9=?(?9 =??;7: @9: >9==9;. Borde del material desplazado.- L??>9 @9: 7>9=?7: @9<:77@ H 7:97@ @9:
•
•
9(7=<9 <=?;(?<7:. Punta.- E>=9 @9: 8=@9 H @?>7;>9 @9 :7 (=;7(?;. Pie.- P=(?; @9 7>9=?7: @9<:77@+ *9 9 7<7 8=9 9: >9==9; =??;7: 7 <7=>?=
•
@9: *=7. Cuerpo principal.- P=(?; @9: 7>9=?7: @9<:77@ 7<7@ 8=9 :7 *<9=?(?9 @9
•
•
=>*=7 9;>=9 * 9=7:9 @9: @9:?7?9;> '@9=9(K ?*?9=@-. Cabecera.- M7>9=?7: <=H(>?(79;>9 0?; ?>*+ 7@7(9;>9 7 :7 <7=>9 *<9=?= @9:
•
9(7=<9 <=?;(?<7:. Superficie original del terreno.- T7:*@ 9?>9;>9 7;>9 @9 (*==?= 9: ??9;>.
•
•
P*9@9 (;?@9=7=9 ( >7: :7 *<9=?(?9 @9 *; 7;>?* @9:?7?9;> 9>78?:?7@. Superficie de rotura.- S*<9=?(?9 7 >=7 @9 :7 (*7: >?9;9 :*7= 9: ??9;>. Superficie de separación.- S*<9=?(?9 *9 :??>7 9: 7>9=?7: @9<:77@ @9: >9==9;
•
=??;7:. Material desplazado.- M77 @9 7>9=?7: ?@7 K7(?7 7*9=7 @9: >7:*@+ @9@9 *
•
<?(?; =??;7: 9; 9: ?. P*9@9 9;(;>=7=9 9; *; 9>7@ @9=7@. Zona de deflación.- $=97 @9;>= @9 :7 (*7:+ 9: 7>9=?7: @9<:77@ 9 9;(*9;>=7 <=
•
@987 @9 :7 *<9=?(?9 =??;7: @9: >9==9;. Zona de acumulación.- $=97 9; :7 *9 9: 7>9=?7: @9<:77@ 9 @?<;9 <=
•
9;(?7 @9 :7 *<9=?(?9 =??;7: @9: >9==9;. E; :7 F?*=7 " 7<=9(?7 :7 <7=>9 @9 *;7 7::7 (?=(*:7= 9; @;@9
P
: H?7 <=*;@?@7@ @9: @9:?7?9;> 9@?@7 <9=<9;@?(*:7=9;>9 7: <:7; @9: >7:*@.
: H?7 :;?>*@ @9: @9:?7?9;> 9@?@7 9; 9: <:7; @9: >7:*@.
!"
: 7:>*=7 @9: @9:?7?9;> 9@?@7 9;>=9 :7 (789(9=7 9:
K=?;>7: @9@9 9: 7;(?7 UNMSM | LIZET GIOVANNA ORE SANTI 9; *;7 9((?; :;?>*@?;7: @9: @9:?7?9;>.
MODELO DE FALLA CIRCULAR
L D P LD
Figura 9. s1uema de una falla circular (VARNS! "#$%).
5. FACTORES CONDICIONANTES Y DESENCADENANTES DE UNA FALLA CIRCULAR P7=7 7;7:?7= 9>9 >?< @9 7::7 9 @9 <=?=@?7: ?<=>7;(?7 9: =9(;(??9;> @9 : 7(>=9 *9 (;@?(?;7; :7 9>78?:?@7@ 7*9:: >= *9 7(>7; ( @99;(7@9;7;>9 @9 : ??9;>. E: (;(??9;> 78 7(>=9 <9=?>9 @9>9=?;7= :7 <?8:9 (*==9;(?7 @9 9>7 7::7 <7=7 7 >7= 9@?@7 ;9(97=?7 <7=7 9?>7=: (==9?=:. L7 <=?;(?<7:9 (;@?(?;9 ; • • • •
L7 9>=*(>*=7 9:?(7. L7 :?>:7. L7 (;@?(?;9 K?@=9:?(7. L7 =:7.
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MODELO DE FALLA CIRCULAR
U;7 7=?7(?; @9 9>7 (;@?(?;9 <*9@9 <=@*(?= *; ?;(=99;> @??;*(?; @9: 9*9= @9 (=>9 (* 99(> ?;9@?7> @99;(77 9; 9: @9:?7?9;> @9 :7 77 <>9;(?7:9;>9 7 7::7=.
. RESISTENCIA
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ALTAMENTE FRACTURADOS E; *(K (7 =97:9+ ( 9; 9: <=99;>9 >=787+ K78:7 @9 :7 7::7 (?=(*:7= 9; 9: 7(? =( 7:>79;>9 =7(>*=7@. E; 9> (7 8=9>@ (*7;@ ; K7 @9>9=?;7@7 *<9=?(?9 <=9@?;7;>9 *9 7=9(7; 9: @9:?7?9;>+ : H =7;78:9+ ( <=?9=7 8*9;7 7<=?7(?;+ 9 9: * @9 :7 >9=7 @9 Mecánica de Suelos <7=7 9: 7;H:?? @9: (;*;>+ *<;9= *9 9 >=7>7 =97:9;>9 @9 *; *9:. T78?; 9 <?8:9+ * 7<:?(7(?; <=H(>?(7 9 ??:7=+ *>?:?7= : ndices de !alidad *9 9 <*9@9; 7?;7= 7: 7(? =( 7:=7= :7 =9?>9;(?7 @9: (;*;> 9; 879 7 : (=?>9=? *7@ 9; 7(? =( 9:78=7@ <= OE Y BRON <= BIENIASI. O>=7 >9=(9=7 <?8?:?@7@ 9 :7 @9 *>?:?7= e"#rapolar las #eorías @9 LADANYI ARCAMBAULT+ :7 @9 BARTON. E; (*7:*?9=7 @9 : (7 9: (;*;> 9 7;7:?7 ( ? @9 *; 9@? K;9 9 >=7>79+ (; * (K9?; H;*: @9 =7?9;> ?;>9=; (==9<;@?9;>9. L (;(9<> 8H?( @9 :7 =9?>9;(?7 7: (=>9 9 *<;9; (;(?@ '?> 9; 9: (?(: 7;>9=?=-+ >7; : 9 =9(=@7=7 *9 <7=7 * @9?;?(?; *9:9 9<:97=9 9: (=?>9=? @9 MK= C*:8+ *9 9; * =7 H 9;9=7: 9; >=?; @9 >9;?;9 99(>?7 9 7*>7 7 :7 ?*?9;>9 9<=9?; τ =c + ( σ −u ) tan ϕ = c + σ'tanϕ ' '
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EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
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MODELO DE FALLA CIRCULAR
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<9=<9;@?(*:7=9;>9 7 :7 *<9=?(?9 @9 @9:?7?9;>+ σ ' 9 <= >7;> :7 >9;?; 99(>?7 (==9<;@?9;>9. T78?; 9 <=9(? >9;9= 9; (*9;>7 *9 :7 ?:?7(?; @9 :7 =9?>9;(?7 7: (=>9 @9: >9==9; <*9@9 9>7= *9>7 7 7=(7@7 7=?7(?;9 9; *;(?; @9: ;?9: @9 @9=7(?;9. E>7 (?=(*;>7;(?7 <*9@9; @7=9 9; 7=(?::7 @9 9:97@7 <:7>?(?@7@+ 9<9(?7:9;>9 9; :7 8=9(;:?@7@7+ 9; :7 *9 9 =9(*9;>9 9;(;>=7= @?9=9;(?7 *>7;(?7:9 9;>=9 :7 =9?>9;(?7 H?7 @9 0
Figura 1$. Resistencia al corte de 2pico3 y 2residual3! 4ndice de 5ra'ilidad.
Figura 11. 6n'ulos de rozamiento en función del contenido de arcilla.
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MODELO DE FALLA CIRCULAR
Figura 12. Movilización de resistencia al corte en diversos puntos de una misma superficie de deslizamiento.
!. FACTOR DE SEGURIDAD E: 7;H:?? @9 : 9(7;? @9 7::7 9; : >7:*@9 ; <9=?>?=H (*7;>??(7= 9: 7(>= @9 9*=?@7@+ 9: *9 7 * 9 9;>=97 :7 (;@?(?; ;?7 9*=7 <7=7 :7 ?;(:?;7(?; @9 :7 <7=9@ @9: >7:*@. 'F?*=7 13-
Figura 13. 5actor de se'uridad en taludes estables e inestables.
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MODELO DE FALLA CIRCULAR
!.1
FACTOR DE SEGURIDAD DEFINIDO COMO
RELACIÓN DE MOMENTOS E: (9?(?9;>9 @9 9*=?@7@ @9 *; >7:*@ K7 @9 =9<=99;>7= :7 =9:7(?; 9?>9;>9 9;>=9 7((?;9 9>78?:?7@=7 '=9?>9;>9- *9=7 @99>78?:?7@=7. 'F?*=7 14-. E;>=9 :7 7((?;9 9>78?:?7@=7 9 9;(*9;>=7 :7 =9?>9;(?7 7: (=>9 @9: >9==9;+ :7 =9?>9;(?7 @9 :7 9>=*(>*=7 @9 *9(?;+ :7 *9=7 9>78?:?7@=7 9>9=;7+ : <9 9>78?:?7@=9+ 9>(. E;>=9 : 9:99;> >9;@9;>9 7 7=9(9= :7 ?;9>78?:?@7@ ; :7 =7?>7>=?7 <9 @99>78?:?7@=9+ :7 <=9?;9 @9 7*7 ?;>9=>?(?7: :7 *9=7 @99>78?:?7@=7 9>9=;7. E; >=?; 9;9=7:9 9: (9?(?9;>9 @9 9*=?@7@ F 9 <*9@9 @9?;?= ( 9: ((?9;>9 9;>=9 9: 9;> @9 :7 *9=7 =9?>9;>9 7: @9:?7?9;> 7 : :7= @9 :7 :;97 @9 =>*=7 '9>78?:?7@=7- 9: 9;> @9 :7 *9=7 *9 >?9;@9; 7 9= :7 77 ?;9>78:9 '@99>78?:?7@=7-.
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MODELO DE FALLA CIRCULAR
Figura 14. Acciones estabilizadoras y desestabilizadoras en un talud con supuesta D9 9;>=9 : 9:99;> 7;>9=?=9+ 9; 9;9=7: :7 <=9?;9 ?;>9=>?(?7:9 :7 =9?>9;(?7 7: (=>9 @9: >9==9; ; :7 7=?78:9 *9 H ?;:*9; 9; :7 9>78?:?@7@+ <= 9:: 9 EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
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MODELO DE FALLA CIRCULAR
*;@79;>7: (;(9= :7 (;@?(?;9 H @97=78:9 @9 <=9?; ?;>9=>?(?7: *9 <*9@7; @7=9+ 7 ( 9>?7= :7 =9?>9;(?7 7: (=>9 @9: >9==9; @?<;?8:9 7 : :7= @9 (*7:*?9= *<9=?(?9 <>9;(?7: @9 @9:?7?9;>.
!.2
FACTOR DE SEGURIDAD DEFINIDO COMO UNA
REDUCCIÓN DE LA RESITENCIA AL CORTE LÍMITE DEL TERRENO E>7 7(9<(?; 9 :7 K78?>*7: 9; 9: (7 @9 >7:*@9 9; *9:. A+ <7=>9 @9: (=?>9=? @9 MK= C*:8 9: (9?(?9;>9 @9 9*=?@7@ <*9@9 @9?;?=9 ( :7 =9:7(?; 9;>=9 :7 =9?>9;(?7 7: (=>9 H?7 @?<;?8:9 9; 9: >9==9; ' - 7 : :7= @9 :7 *<9=?(?9 @9 @9:?7?9;> 9:9?@7 :7 9>=?(>79;>9 ;9(97=?7 ' - ?:?7@7 <7=7 (;9*?= 9: 9*?:?8=? 9>H>?( @9: 9(7;? (;?@9=7@. 'F?*=7 15-
Figura 15. sfuerzo cortante movilizado. D;@9 ( ф =9<=99;>7; :7 (K9?; 99(>?7 H;*: @9 =7?9;> ?;>9=; 99(>? ?:?7@ <7=7 7:(7;7= @?(K 9*?:?8=?.
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MODELO DE FALLA CIRCULAR
". M#TODOS DE CÁLCULO L >@ @9 (H:(*: <7=7 7;7:?7= :7 9>78?:?@7@ @9 *; >7:*@ 8*(7; @9>9=?;7= 9: 7(>= @9 9*=?@7@ 'FS- @9: ?+ 9 @9(?= 9: =7@ @9 9*=?@7@ @9 *; 7(? =( =9;>9 7 *;7 <?8:9 7::7 (?=(*:7=. 'F?*=7 16E?>9; 7=? >@ @9 7;H:?? @?<;?8:9 <7=7 (7:(*:7= :7 <?8?:?@7@ @9 *9 9 @97==::9 *; @9:?7?9;> @9 >?< (?=(*:7= 9; 9: (*9=< @9: >7:*@+ (*7 @9(=?<(?; (;>?>*9 9: 89> @9 9>9 >=787.
DEFINICIÓN DE LOS M#TODOS DE CÁLCULO M#$%"%S "E E&'((B)(% (M($E. S9 877; 9(:*?79;>9 9; :7 :99 @9 :7 9>H>?(7 <7=7 @9>9=?;7= 9: 9>7@ @9 9*?:?8=? @9 *;7 77 <>9;(?7:9;>9 ?;9>78:9. N >?9;9; 9; (*9;>7 :7 @9=7(?;9. S*<;9; *9 :7 =9?>9;(?7 7: (=>9 9 ?:?7 >>7: ?*:>H;979;>9 7 : :7= @9 :7 *<9=?(?9 @9 (=>9. A * 9 9 <*9@9; (:7??(7= 9; @ =*<
M*todos e+actos. L7 7<:?(7(?; @9 :7 :99 @9 :7 9>H>?(7 <=<=(?;7 *;7 :*(?; 97(>7 @9: <=8:97 (; :7 ;?(7 7:9@7@ @9 :7 ?<:??(7(?;9 <=@ : >@ @9 9*?:?8=? :??>9 '7*9;(?7 @9 @9=7(?;9+ 7(>= @9 9*=?@7@ (;>7;>9 9; >@7 :7 *<9=?(?9 @9 =>*=7-. E> : 9 <?8:9 9; (7 @9 99>=7 9;(?::7 ( <= 99<: :7 7::7 <:7;7= 7::7 (?=(*:7=. M*todos no e+actos.- E; :7 7= <7=>9 @9 : (7 :7 99>=7 @9 :7 *<9=?(?9 @9 =>*=7 ; <9=?>9 8>9;9= *;7 :*(?; 97(>7 @9: <=8:97 9@?7;>9 :7 ;?(7 7<:?(7(?; @9 :7 9(*7(?;9 @9 :7 9>H>?(7. E: <=8:97 9 hiperes#á#ico K7 @9 K7(9=9 7:*;7 ?<:??(7(?; K?<>9? <=9?7 *9 <9=?>7 * =9:*(?;. D??@?;@9 >78?; 9; @ =*<
E,uilibrio global de la masa deslizante.- <=H(>?(79;>9 9; @9*. D9;>= @9 9:: 9;(;>=7
M*todo de anlisis no drenado.- S9 (;?@9=7 *; H;*: @9 =7?9;> ;*:.
EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
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MODELO DE FALLA CIRCULAR
M*todo de anlisis general drenado/: C0rculo de fricción o rozamiento. S9 =97:?7 K?<>9? =9<9(> 7 :7 >9;?;9 ;=7:9 9; :7 *<9=?(?9 @9 @9:?7?9;>-. Sistematización del m*todo del circulo de rozamiento: Soluciones
mediante bacos 1baco de $a2lor 'P7=7 >9==9; K;9 ?; =7?9;>1baco de !unter 3 S4uster 'P7=7 >9==9; ?; =7?9;> =9?>9;(?7 (=9(?9;>9 (; :7 <=*;@?@7@1baco de $a2lor 'P7=7 >9==9; K;9 (; (K9?; =7?9;> 1bacos de !oe5 2 Bra2
M*todo de do6elas o fa7as 6erticales.- C;?@9=7; 7 :7 77 @9:?7;>9 @??@?@7 9; *;7 9=?9 @9 77 9=>?(7:9. 'S9 =97:?7 K?<>9? =9<9(> 7 :7 *9=7 :7>9=7:9 9;>=9 :7 @9:7-. E?>9 *;7 =7; 7=?9@7@ @9 >@ *9 (;?@9=7; @?9=7 K?<>9?. C:7??(H;@9 9; M#TODOS DE CÁLCULO o
M%#odos apro"i&ados.' N (*<:9; >@7 :7 9(*7(?;9 @9 :7 9>H>?(7. S9 <*9@9; (?>7= ( 99<: : >@ @9
DE F9::9;?*+ J7;8* B?K@7 :7 E$UILIBRIO ES LIMITE 9(*7(?;9 @9 :7 9>7>?(7. L H (;(?@ ; : >@ /M0to,o' nu$0r!"o'1 @9 M=9;>9=;P=?(9+ S<9;(9= B?K< =?*=.
M#$%"%S "E C1C'% "E "EF%)M8C(%9ES.- C;?@9=7; 9; 9: (H:(*: E%ACTOS :7 @9=7(?;9 @9: >9==9; 7@9H @9 :7 :99 @9 :7 9>H>?(7. * 7<:?(7(?; NO E%ACTOS
Rotura Planar Rotura C!r"ular <=H(>?(7 9 @9 =7; (<:9?@7@ 9: <=8:97 @989 Rotura en Cua
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ESTABILIDAD GLOBAL
M#TODOS DE DOVELAS
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MODELO DE FALLA CIRCULAR
Figura 16. lasificación de los m,todos de clculo de estabilidad de taludes.
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MODELO DE FALLA CIRCULAR
L >@ @9 9*?:?8=? :?>9 ; *(K H *>?:?7@ *9 : >@ @9 (H:(*: 9; @9=7(?;9 7 <97= @9 *9 9> <=<=(?;7; *; 7;H:?? *(K H (<:9> @9 :7 9>78?:?@7@ @9: >7:*@. E> @98?@ 7 *9 : >@ @9 (H:(*: 9; @9=7(?;9 ; *(K H :7= (> 9; * 99(*(?;+ *9 : m,todos de e1uilibrio límite estn ampliamente contrastados en la prctica y se conocen sus límites y 'rados de confianza.
!(P$ES(S B1S(C8S 1. S9 (;?@9=7 *9 9: >7:*@ 9 ?;@9?;?@ 9; :7 @?=9((?; K=?;>7: <7=7:9:7 7 * *<9=?(?9+ 9 @9(?=+ 9: <=8:97 9 9>*@?7 9; (;@?(?;9 8?@?9;?;7:9. 2. S9 *<;9 *; 9(7;? @9 =>*=7 7 7= @9 @9>9=?;7@7 *<9=?(?9 @9 @9:?7?9>; '9; 9>9 (7 (*=7-. L7 77 (;>9;?@7 <= @?(K7 *<9=?(?9 9 (;?@9=7 ( *; 8:*9 =??@+ 8?9; 9 *8@??@9 9; 8:*9 7 <9*9Q+ >78?9; =??@. 'F?*=7 1-.
Figura 17. ompartimentacion en blo1ues en los metodos de e1uilibrio limite. 3. S9 7*9 *; (=?>9=? @9 =>*=7 @9: >9==9; 7 : :7= @9 :7 *<9=?(?9 @9 @9:?7?9;> @9?;?@7. D?(K (=?>9=? 9 K78?>*7:9;>9 9: @9 MK=C*:8+ 7 ?> :?;97 7==?87. 4. S9 *<;9 *9 (7@7 8:*9 9; : *9 9 K7 *8@??@?@ :7 77 9 9;(*9;>=7 9; 9*?:?8=? 9>=?(>. 'F?*=7 1!-. 5 S9 @9?;9 *; 7(>= @9 9*=?@7@ *9 9 *<;9 constante a lo largo de toda la
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MODELO DE FALLA CIRCULAR
Figura 18. 5uerzas 1ue actuan en una rotura circular. D;@9 • • • • •
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• •
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EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
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MODELO DE FALLA CIRCULAR
S9 <*9@9 @?>?;*?= 7* 9;>=9 : >@ *9 (;?@9=7; 9: e(uili)rio glo)al de la &asa desli*an#e+ <=H(>?(79;>9 9; @9*+ : &%#odos de do+elas+ 77 9=>?(7:9. E>9 :>? 9 9; 9: *9 (7? :7 >>7:?@7@ @9 : >@ *>?:?7@ 7(>*7:9;>9 9 877; (;?>9; 9; @??@?= 9: 7(? <>9;(?7:9;>9 @9:?7;>9 9; =987;7@7 9=>?(7:9+ (7:(*:7= 9: 9*?:?8=? @9 (7@7 *;7 @9 9::7+ ?;7:9;>9 7;7:?7= 9: 9*?:?8=? :87:+ 8>9;?9;@ 9: 7(>= @9 9*=?@7@+ *9 9 @9?;9 ( :7 =9:7(?; 9;>=9 *9=7 9;> =9?>9;>9 *9=7 9;> >=9.
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E$UILIBRIO GLOBAL DE LA MASA.
S; : >@ H 7;>?* ; H:?@ >7; : <7=7 *9: K;9 7 *9 9 *<;9; (;>7;>9 : <7=H9>= =9?>9;>9 9; >@7 :7 77 *9 @9:?7. P7=7 7;7:?7= :7 9>78?:?@7@ @9 *; >7:*@ @9 (7=7(>9=>?(7 =9?>9;>9 99>=7 @9>9=?;7@7+ 9 ;9(97=? (;(9= 9: (9;>= 9: =7@? @9: (=(*: <= @;@9 9 <=@*(9 9: @9:?7?9;>. E>9 K7 @9 7>?7(9= :7 (;@?(?; @9 *9 :7 =9:7(?; 9;>=9 :7 =9?>9;(?7 7: (=>9 @9: 7(? =( 7 : :7= @9 :7 *<9=?(?9 :9 9*9= >7;9;(?7:9 97 :7 ;?7 @9 >@7 :7 *<9=?(?9 <?8:9. S* <?(?; 9 *9:9 9>?7 9@?7;>9 >7;>9. E: <=8:97 9 9 =9:97@ 9; :7 ?*=7 1!. E; @;@9 9 (;(?@ 9: <9 9<9(?( @9: 7(? =( :7 99>=7.. E; 9;9=7: 9 (;(9; 9: <9 @9 :7 77 *9 @9:?7 + :7 =9*:>7;>9 @9 :7 *9=7 9>9=?=9 A+ >78?; :7 @?>=?8*(?; @9 <=9?; ?;>9=>?(?7: <@=H 9;(;>=7=9 * =9*:>7;>9 U. P7=7 (;(9= T( K7(9 7:>7 F. P7=7 (;(9= T, K7(9 7:>7 (;(9= :7 =9*:>7;>9 @9 :7 *9=7 ;=7:9 99(>?7 *9 7(>7; 8=9 :7 *<9=?(?9 @9 @9:?7?9;> N+ F = ф. D9 N : 9 789 *9 <77 <= 9: (9;>= @9: (=(*: @9(;(?;@9 * 7;?>*@ 9: >= <7=H9>= @?=9((?;7:. A? <*9 9 (*9;>7 (; 4 ?;(;?>7 'F+ 7;?>*@ @9 N+ <7=H9>= @9 :7 :;97 @9 7((?; N+ = ф- : >=9 9(*7(?;9 '<=9((?; 9; /+ <=9((?; 9; Y 9*?:?8=? @9 9;>-. A? <*9 9: <=8:97 9 9>H>?(79;>9 ?;@9>9=?;7@ 9 ;9(97=? =97:?7= K?<>9?. &.1.1 ANÁLISIS NO DRENADO '()C*+ ,)-
C;?@9=7 *9 9: H;*: @9 =7?9;> @9: 7(? =( 97 ;*:. S? 9 >7; 9;> =9<9(> 7: (9;>= @9: (=(*:+ >7;> N ( U @7; 9;> ;*:. E:
EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
23
MODELO DE FALLA CIRCULAR
9;> =??;7@ <= 9: <9 9 H(?: @9 (7:(*:7= 9: 9;> M T =??;7@ <= :7 *9=7 >7;9;(?7:9 9=H 'S? :7 =9?>9;(?7 ; @=9;7@7 C* 9 (;>7;>9- M T =
C u F
^
. r . AB
C u
D;@9 F 9 :7 >9;?; @9 (=>9 ?:?7@7 'F ) F7(>= @9 9*=?@7@-. O8:?7;@ 7 :7 9?>9;(?7 @9 9*?:?8=? @9 9;> 9 8>?9;9 9: 7(>= @9 9*=?@7@
^
C u .r . AB F = M W
D;@9 M W =W . d S? 9: *9: <=99;>7 7=?7(?;9 9<7(?7:9 @9 C* 'U*7:9;>9 C* 7=?7 :?;97:9;>9 (; :7 <=*;@?@7@- 9: 7(>= 9 9*=?@7@ <*9@9 9= (7:(*:7@ ( B
∫ C .r.ds u
F =
A
M W
S? 9: *9: 9 K;9+ :7 :;97 @9 7((?; @9 T( 9=H <7=7:9:7 7 :7 =9(>7 *9 *;9 A B :7 @?>7;(?7 7: (9;>= O 9=H '
r=
^
C u . AB
´ AB
&.1.2 ANÁLISIS GENERAL 'DRENADO/ CÍRCULO DE ROZAMIENTO
S? 9?>9 =7?9;>+ >7;> :7 =9*:>7;>9 @9 :7 *9=7 >7;9;(?7:9 ?:?7@7 'T- ( :7 =9?>9;(?7 *9 <*9@9 <=<=(?;7= 9: *9: 'R-+ <*9@9; @9(<;9=9 9; @ >=?; 'T( Tф- 'R ( R ф- 7(?7@ (7@7 *; @9 9:: 7 :7 *9=7 @9 (K9?; =7?9;> =9<9(>?79;>9. S9 (*<:?=H T =T c + T ф=
Rc Rф
+
F c F ф
= R
F
EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
24
MODELO DE FALLA CIRCULAR
D;@9 Rc Rф ; :7 *9=7 >7;9;(?7:9 =9?>9;>9 (K9?7 =?((?;7: *9 9: 7(? <*9@9 @97==::7= 7 : :7= @9 :7 :;97 @9 =>*=7+ 5 9: 7(>= @9 9*=?@7@ @9 :7 77 @9:?7;>9. Rc
9 >>7:9;>9 (;(?@7 >7;> 9; 7;?>*@ ( 9; @?=9((?;+ 7 *9 *<;?9;@ *9
:7 (K9?; c 9 (;>7;>9 (;(?@7 7 : :7= @9 >@ 9: 7=( @9 @9:?7?9;> @9@9 7 K7>7 8 =9*:>7 cuerda
Rc =c . L AB
cuerda D;@9 L AB 9 :7 7;?>*@ @9 :7 (*9=@7 AB 7@9H 9: 9(>= >?9;9 :7 @?=9((?; @9
@?(K7 (*9=@7. P7=7 @9>9=?;7= :7 @?>7;(?7 Rc 7: (9;>= @9: (=(*: 9 <*9@9 @9>=7= *9 aerco
Rc . r c = c . L AB . r
<= >7;> arco
rc =
Rф
c . L AB . r cuerda
c . L AB
=
arco
L AB . r cuerda
L AB
; 9 (;(?@7 ;? 9; @?=9((?;+ ;? 9; 7;?>*@+ <9= 7 :?7@7 7 N <= '
Rф = N .tanф
<= @9?;?(?; 9 <9=<9;@?(*:7= 7 :7 :;97 @9 7((?; N+ @9 :7 *9 9 789 *9 <77 ;9(97=?79;>9 <= 9: (9;>= @9: (?=(*: @9 =>*=7. P= (;9;?9;(?7 9 (;?@9=7=7 *9 9: 7(>= @9 9*=?@7@ 7(?7@ 7 :7 *9=7 @9 =7?9;> :7 *9=7 @9 (K9?; (?;(?@9; ' F = F c = F ф -. S9 (;(9; + A+ U+ *;> (; :7 @?=9((?; @9 :7 =9?>9;(?7 7: (=>9 (K9?7 8=9 :7 *<9=?(?9 @9 =>*=7 *9 9=H <7=7:9:7 7 :7 (*9=@7 0AB. S?; 987= *9@7; :7 ?*?9;>9 ?;(;?>7 <7=7 =9:9= (<:9>79;>9 9: <=8:97
V7:= @9 F V7:= @9 N
EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
25
MODELO DE FALLA CIRCULAR
U; <7=H9>= <7=7 @9?;?= :7 :;97 @9 7((?; @9 N
U; <7=H9>= <7=7 @9?;?= :7 :;97 @9 7((?; @9 T ф
P= >=7 <7=>9+ : 9 @?<;9 @9 3 9(*7(?;9 @9 9*?:?8=? <7=7 =9:9= 9: <=8:97. P7=7 :*(?;7=: 9 ;9(97=? K7(9= K?<>9? 7(9=(7 @9 :7 =7 @9 :7 @?>=?8*(?; @9 N 7 : :7= @9 :7 *<9=?(?9 @9 =>*=7. E; 9: >@ =??;7: 9 K7(9 :7 K?<>9? @9 *9 #odas las #ensiones nor&ales efec#i+as (ue ac#,an so)re el círculo es#án concen#radas en un pun#o - desconocido/ del &is&o. &asos7 '1- S9 8>?9;9 9: 9(>= D+ =9*:>7;>9 @9: <9 + @9 :7 *9=7 @9: 7*7 U @9 :7 *9=7 9>9=;7 A. 'F?*=7 1"-
Figura 19. 8btención de la resultante *. '2- C;(?@ D 9 9;(*9;>=7 9: <*;> f <= @;@9 @989 <77= :7 =9*:>7;>9 @9 :7 *9=7 N T ф + @9(;(?@7 K7>7 9: 9;>. 'F?*=7 2&'3- D9 ' Rф = N tanф = T ф . F ф T ф
tanф = tan фd N ' F ф
=
S9 @9@*(9 *9 :7 =9*:>7;>9 @9 N T ф =7 *; H;*: @9 ф d (; :7 ;=7: 7 :7 (?=(*;9=9;(?7 @9 =>*=7 9; 9: <*;> /+ : (*7: 9 9*?7:9;>9 7 @9(?= *9 @?(K7 =9*:>7;>9 @989 9= >7;9;>9 7 *; (?=(*:+ ::77@ círculo de ro*a&ien#o0 @9 (9;>= O @9 =7@? r senфd . C;(?@ 9: <*;> @9 <7 f 9 ?<;?9;@ 9>7 :>?7
EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
26
MODELO DE FALLA CIRCULAR
(;@?(?;+ (;@?(?;+ <*9@9 (;(9=9 :7 :;97 @9 7(>*7(?; @9 :7 =9*:>7;>9 =9*:>7;>9 @9 N T ф + 9: 7:= @9 9>7 9: 7:= @9 T c . 'F?*=7 21'4-
EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
27
MODELO DE FALLA CIRCULAR
O B U = D
U
A
L;97 @9 7((?; T (
Figura2$. 8btención del punto f
C=(*: @9 =7?9;>
= 9; ф@
B
T" Tф
N /
A
NTф
D
L;97 @9 7((?; T (
Figura 21. *efinición 21. *efinición del círculo de rozamiento.
C;; 9: 7:= C 7:= 8>9;? 8>9;?@ @ @9
T c
<@=9 K7::7= 9: 7:= @9 F 9@?7;>9 :7 ?*?9;>9
9<=9?;
EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
28
MODELO DE FALLA CIRCULAR
R c F = T c
L7 :*(?; 9 K78=H 9;(;>=7@ ? F c
F ф (?;(?@9;. C 9; 9;9=7: 9> ;
(*==?=H+ 9 9;77=7 *; ;*9 F ф ': *9 9 @9?;9 ( (?=(*: @9 =7?9;>- 9 8>9;@=H *; ;*9 F c . C*7;@ 9 K77; 9;(;>=7@ 7=? 7:=9 @9 F c F ф '3 9 **7:9;>9 *?(?9;>9- 9 K7(9 *; =H?( 7;H: 7: @9 :7 ?*=7 22+ 9>7 (;?>9 9; >=77= *;7 (*=7 @9 <*;> ' F c F ф -+ * ?;>9=9((?; ?;>9=9((?; (; :7 =9(>7 F c ) F ф '7 45W @9@9 9: =?9; @9 (=@9;7@7- <=<=(?;7 9: 7(>= @9 9*=?@7@ 8*(7@. F c
F c = F ф
F = F c = F ф
P*;> 8>9;?@ 9; @?9= >7;>9
F ф
Figura 22. 8btención 22. 8btención del factor de se'uridad! solución del problema. E?@9;>99;>9 <*9@9; 7;7:?7=9 >=7 K?<>9? 7(9=(7 @9 :7 @?>=?8*(?; @9 N. F= :?(K (;?@9=7 :7 K?<>9? @9 *9 las #ensiones nor&ales efec#i+as es#%n concen#radas en los dos pun#os e"#re&os de la superficie de ro#ura 8 2 B/.
9%$8: L <7 7;>9=?=9 <9=?>9; @9>9=?;7= 9: 7(>= @9 9*=?@7@ @9 un círculo de desli*a desli*a&ien# &ien#oo de#er&in de#er&inado. ado. P7=7 P7=7 9>?7 9>?7== 9: general del #alud 9=H ;9(97=? =9<9>?= 9: <=(9 (<:9> (; >=7 *<9=?(?9 (?=(*:7=9 @9 @9:?7?9;> K7>7 8>9;9= :7 H @97=78:9+ 9 @9(?=+ :7 @9 (9?(?9;>9 @9 9*=?@7@ ;?.
EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
29
MODELO DE FALLA CIRCULAR
L7 K?<>9? @9 *<;9= N (;(9;>=7@7 9; *; <*;>+ <=<=(?;7 : (9?(?9;>9 @9 9*=?@7@ 9*=?@7@ H 87 @9 : *9 9 8>?9;9; 8>?9;9; 9<:97;@ @?9=9;>9 K?<>9? K?<>9? @9 @?>=?8*(?; @?>=?8*(?; @9 N+ ?9;>=7 *9 :7 K?<>9? @9 F= :?(K (;@*(9 7 : (9?(?9;>9 @9 9*=?@7@ H 7:>. S9 @?(9 9> 7 *9 9 K7 @9>=7@ @9>=7@ *9 9: 7(>= @9 9*=?@7@ 'F- =97: @9 *;7 7::7 (?=(*:7= 9>H *(K H <=? 7: :?>9 ?;9=?= *9 7: *<9=?=+ <= : *9 9: >@ @9 (?=(*: @9 =7?9;>+ 7*;*9 (;9=7@=+ =9*:>7 7<:?(78:9 9; :7 ?;9;?9=7 <=H(>?(7. &.1.3 SISTEMA SISTEMATIZACIÓN TIZACIÓN DEL M#TODO M#TODO DE CÍRCULO CÍRCULO DE ROZAMIENTO/ ROZAMIENTO/ SOLUCIONES MEDIANTE ÁBACOS
E; : : > >@ @ ?> ?> 7;>9 7;>9=? =?= =9; 9;>9 >9 9 K7 @9(= @9(=?> ?> ( ( 8 8>9 >9;9= ;9= 9: (9 (9?( ?(?9 ?9;> ;>99 @9 9*=?@7@ 9*=?@7@ @9 *;7 @9>9=?;7@7 *<9=?(?9 *<9=?(?9 @9 @9:?7?9;> @9:?7?9;> 9>78:9(?9;@ 9>78:9(?9;@ :7 9(*7(?;9 9(*7(?;9 @9 9*?:?8=? @9 :7 77 @9 ?;:*(=7@7 =9?9;@ 9: ? <=(9@??9;> @9 (H:(*: K7>7 8>9;9= :7 H @97=78:9 ':7 *9 <=<=(?;7 9;= (9?(?9;>9 @9 9*=?@7@-+ : *9 ?; @*@7 =9*:>7 87>7;>9 >9@? :9;>. F=9;>9 7 9::+ @?9= 7*>=9 9; 9: >=7;(*= @9 :7 K?>=?7 (;(9;>=7=; * 9*9= 9; :7 8>9;(?; @9 H87( @9 * H 9;(?:: 9 ?;>9 ?;>9= =9@ 9@? ?.. E? E?>9 >9;; =7; =7; (7;> (7;>?@ ?@7@ 7@ @9 H87( H87( <7=7 <7=7 :7 =7; =7; 7=? 7=?9@ 9@7@ 7@ @9 7=? 7=?78 78:9 :9+ + @9(=?8?=9 7:*; @9 : @9 * H (; <=7(>?(. P7=7 9: 9<:9 @9 : H87( 9; 9;9=7:+ 9 *7 *;7 @9>9=?;7@7 ;9;(:7>*=7 *9 <9=?>9 @?>?;*?= : @?9=9;>9 >?< @9 (=(*: @9 @9:?7?9;> ?>7 9; 9: <*;> 4.3 @9: <=99;>9 >=787.
1B8C% "E $8;%) P8)8 $E))E9%S !%M%78?:?@7@ @9 >7:*@9 9@?7;>9 9: >@ @9 TAYLOR '1"3- 9 877 9; 9: &%#odo del círculo de ro*a&ien#o ro*a&ien#o + : *9 *<;9 (;?@9=7= *9 :7 >9;?;9 ;=7:9 9; :7 *<9=?(?9 @9 @9:?7?9;> 9 (;(9;>=7; 9; *; ;?( <*;>. E: 7;H:?? 7;H:?? 9 K7(9 9; <=9?;9 >>7:9+ >>7:9+ <= : *9 :7 (K9?; c 9: H;*: @9 =7?9;> ?;>9=; , K7; @9 (7:(*:7=9 9; <=9?;9 >>7:9+ 7@<>7;@ * 7:=9 99(>? : (*7;@ 9: >9==9; 9>H 9(+ (7 <7=7 9: *9 : H87( @9 TAYLOR >?9;9; H *>?:?@7@ 9; >9==9; >9==9; (; (K9?; H;*: @9 =7?9;> ?;>9=;. ?;>9=;. S; >78?; >78?; >?:9 <7=7 (H:(*: (H:(*: @9 9>78?:?@7@ 7 (=> <:7 9; *9: (K9? '(;@?(?; , ) &-. S9 (;?@9=7 9: >9==9; K;9+ 9> 9+ :7 (K9?;+ 9: H;*: @9 =7?9;> ?;>9=; :7 @9;?@7@ ; (;>7;>9 9; >@7 :7 77 @9: >9==9;.
EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
30
MODELO DE FALLA CIRCULAR
E: (9?(?9;>9 @9 9>78?:?@7@ 9 7@?9;?;7: @9<9;@9 ;?(79;>9 @9: H;*: @9 >7:*@ X @9: H;*: @9 =7?9;> ?;>9=; ,. E; 9: H87( =9<=99;>7@ 9; :7 ?*=7 22+ 7<7=9(9; =9:7(?;7@ N s + X , <7=7 7:=9 @9 , 9;>=9 &W 7 25W 7:=9 @9 X @9 &W "&W.
Figura 23. Relación entre N s ! 9 y :. (0aylor)
C*7;@ 9: H;*: @9 =7?9;> ?;>9=; ф 7:9 &W 9 <?8:9 9: 7;H:?? >9;?9;@ 9; (*9;>7 :7 9?>9;(?7 7 (?9=>7 <=*;@?@7@ @9 *;7 (7<7 @*=7 *9 ?<;7 *; :?>9 7 :7 <=*;@?@7@ 7:(7;78:9 <= : (=(*: @9 @9:?7?9;> '=>*=7 <= :7 879-. P7=7 9>9 (7 9 <*9@9 7<:?(7= 9: H87( @9 :7 ?*=7 25+ 9; :7 *9 9 =9:7(?;7; N s + X D+ ?9;@ D 9: coeficien#e o fac#or de profundidad0 @9?;?@ ( :7 =9:7(?; 9;>=9 <=*;@?@7@ @9 :7 (7<7 @*=7+ 9@?@7 @9@9 9: :?>9 *<9=?= @9: >7:*@+ :7 7:>*=7 @9 >9. E; :7 ?*=7 25+ 9 7<=9(?7 *9 : (=(*: H @97=78:9 <*9@9; 9= @9 >=9 >?< @?9=9;>9
!írculo de pie (=(*: *9 <77; <= 9: 7:*@. !írculos de #alud (=(*: *9 (=>7; 7: >7:*@.
EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
31
MODELO DE FALLA CIRCULAR
!írculos de )ase o de pun#o &edio ; (=(*: >7;9;>9 7 :7 879 ?=9 (* (9;>= 9>H 8=9 :7 9=>?(7: *9 <77 <= 9: <*;> 9@? @9: >7:*@.
E; :7 ?*=7 25+ 7@9H @9: (9?(?9;>9 @9 9>78?:?@7@ N s 9 <*9@9 8>9;9= 9: >?< @9 (=(*: H @97=78:9 <7=7 9: (7 (;?@9=7@. E<:?(7=9 :7 *>?:?7(?; @9 9>9 H87( <9= <=9?79;>9 @9?;?=9 7:*;7 7=?78:9 9>=?(7 *7@7 'F?*=7 24- <7=7 *; 9= 9;>9;@??9;>.
Figura 24. *efinición de variables 'eom,tricas para el uso de los bacos de 0aylor. D;@9
•
D
•
:7 (=;7(?; @9: >7:*@+ @9: <*;> H 87 @9: (=(*: (=>?(. / D?>7;(?7 @9: <*;> @9 7:=7?9;> @9: (=(*: (=?>?( 7: 7:*@ 9;
• •
A:>*=7 @9: >7:*@. I;(:?;7(?; @9: >7:*@ F7(>= @9 <=*;@?@7@. E: <=@*(> D 9Q7:7 :7 <=*;@?@7@+ 9@?@7 @9@9
*;(?; @9 :7 7:>*=7 @9: ?. S9 (;?@9=7 <?>? (*7;@ @?(K (=(*: <7=7 <= • •
@987 @9: = (?=(*:7= *9 @9?;9 9: (=(*: (=>?(. $;*: *9 =7 :7 (*9=@7 @9: (=(*: (=>?( (; :7 K=?;>7:.
P= >=7 <7=>9+ 9: * @9 *; ? H87( <7=7 (*7:*?9= >?< @9 >7:*@ =9*?9=9 :7 *>?:?7(?; @9 7:*; <7=H9>= 7@?9;?;7:9. T7:= 9<:9 <7=7 9:: 9: ::77@ !oeficien#e de s#a)ilidad s 0 @9?;?@ (
EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
32
MODELO DE FALLA CIRCULAR
N s =
γ . H c
D;@9 γ P9 9<9(?( 7<7=9;>9 @9: >9==9;.
•
A:>*=7 @9: >7:*@ @9?;?@7 9; :7 ?*=7 24. c CK9?;
•
•
9(K7 9>7 (;?@9=7(?;9+ :7 ?*=7 25 =9<=@*(9 : H87( @9 T7:= <7=7 9: (7 @9(=?> •
'257- R9(9 :7 =9:7(?; 9;>=9 9: H;*: @9: >7:*@ X+ 9: 7(>= @9 <=*;@?@7@ 9:
•
(9?(?9;>9 @9 9>78?:?@7@ N s. '258- M*9>=7 :7 =9:7(?; 9;>=9 9: H;*: @9: >7:*@ X : H;*: *9 ?>7;
•
9: (?=(*: @9 ?( (*7;@ X [ 5&W. '25(- R9:97 :7 =9:7(?; 9;>=9 9: H;*: @9 >7:*@ X 9: 7(>= @9 <=*;@?@7@ 'D <7=7 7=? 7:=9 @9 /.
E; 9: (7 @9 , ) & 9 <*9@9 8>9;9= 9: 7(>= @9 9*=?@7@ @9 =7 @?=9(>7. E;>=7;@ 9; 9: H87( @9 :7 ?*=7 257 (; : 7:=9 @9 'e o%t!ene un alor ,e
N s
'!en,o el :a"tor ,e 'e-ur!,a,; F =
c . N s γ . H
P7=7 *; , < & 9: 7(>= @9 9*=?@7@ 9 K7 @9 8>9;9= 9@?7;>9 >7;>9+ (9;7;@ <= *; 7(>= F1 9 9;>=7 9; 9: H87( @9 :7 ?*=7 257 (; 9: H;*: @9 >7:*@ X+ 9: ?*?9;>9 7:= @9 N s N s =
γ . H . F1 ca
S9 8>9;@=H @9 9>7 =7 *; 7:= ,1 *9 <=<=(?;7 *; ;*9 7(>= @9 9*=?@7@ F2 @7@ <=
EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
33
MODELO DE FALLA CIRCULAR
F 2 =
tan ф tan ф 1
N=7:9;>9 F2 9=H @?>?;> @9 F1. D9<* @9 7:*; >7;>9 9 <*9@9 8>9;9= F = F = F 1
2
3
Figura 25. 6bacos de 0aylor para terrenos sin rozamiento. EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
34
MODELO DE FALLA CIRCULAR
C 9;(?;7 :;97 7==?87+ : H87( @9 T7:= >78?; 9 *>?:?7; <7=7 (7:(*:7= 9: 7(>= @9 9*=?@7@ @9 >7:*@9 7 (=> <:7 *<;?9;@ 7:=9 @9 c (;>7;>9. E; :7 ?*=7 267 268 T7:= <=<=(?;7 9: 7(>= @9 E>78?:?@7@ C d / γ . H 9; *;(?; @9: H;*: @9 >7:*@+ i! @9 :7 <=*;@?@7@ @9 *;7 (7<7 =?@7. C d 9 :7 =9?>9;(?7 7: (=>9 ?:?7@7 C d=C u / F . P= 9:: 9> =H?( <*9@9; *7=9 <7=7 (7:(*:7= *; @9 : 7(>=9 @9>9=?;7;>9 @9 :7 9>78?:?@7@ ' 5! C u ! ; o i) 9; *;(?; @9 : @9H.
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MODELO DE FALLA CIRCULAR
a
EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
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MODELO DE FALLA CIRCULAR
) Figura 26. (0aylor "#?%)
1B8C% "E !'9$E) 3 SC!'S$E) P8)8 $E))E9%S S(9 )%Z8M(E9$% ; )ES(S$E9C(8 C)EC(E9$E C%9 8 P)%F'9"("8" E * K78?>*7: *9 :7 =9?>9;(?7 7: (=>9 ?; @=9;79 ; 97 (;>7;>9+ ?; *9 7*9;>9 (; :7 <=*;@?@7@ '9; *;7 7=(?::7 ;=7:9;>9 (;:?@7@7+ 9; >9=7 @9 =7 :?;97:-. E?@9;>99;>9+ ? 9 K7 @9 @?9Q7= *;7 9(77(?; 9; 9>9 >?< @9 *9: <*9@9 9= ?;>9=97;>9 >9;9= 9; (*9;>7 9>9 99(>. E: <=(9 7 9*?= 9 9: ?*?9;>9
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MODELO DE FALLA CIRCULAR
•
S? :7 =9?>9;(?7 7: (=>9 ?; @=9;79+ S u ; 9 ;*:7 9; :7 (=;7(?; @9: >7:*@+ 9 9>=7<:7 9: <9=?: :?;97: @9 =9?>9;(?7 K7>7 @9>9=?;7= 9: 7:= @9 H 0 9; 9: *9 Su
•
•
9=?7 ;*:. 'F?*=7 27-
H 0 M = S9 (7:(*:7 9: 7(>= H + ?9;@ :7 7:>*=7 @9: >7:*@.
E; *;(?; @9 :7 ?;(:?;7(?; @9: >7:*@ @9 M 9 @9>9=?;7 9: ,&ero de es#a)ilidad @9: H87( @9 :7 F?*=7 28.
•
b A <7=>?= @9 :7 =9?>9;(?7 7: (=>9 ?; @=9;79 7: ;?9: @9: 7:*@+ S u
9
8>?9;9 9: 7:= @9: 7(>= @9 9*=?@7@ F+ *9 ?9;9 @7@ <= :7 9<=9?; b
Su F = N γ . ( H + H 0)
D;@9 γ 9 9: <9 9<9(?( 7<7=9;>9 @9: *9:+ <=9@?7@ ? 7=?7 7 : :7= @9 :7 9=>?(7: @9: >7:*@.
Figura 27. 6bacos de ;unter - Scuster ("#@%) EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
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MODELO DE FALLA CIRCULAR
1B8C% "E $8;%) P8)8 $E))E9%S !%M%<#9E%S C%9 C%!ES(9 ; )%Z8M(E9$% E>9 (7 9 9;(*9;>=7 =9*9:> 9; 9: H87( @9 :7 F?*=7 2+ ? 8?9; * 7:?@9 9 :??>7@7 7 *9 : (;?@9=7 9: >7:*@ 09(+ ; <=9?;9 ?;>9=>?(?7:9. L7 ;9;(:7>*=7 9<:97@7 9 ??:7= 7 :7 @9: H87( @9 T7:= 7;>9 ?>+ (; :7 ?*?9;>9 89=7(?;9 •
ф7 9 9: H;*: @9 =7?9;> 99(>? ?:?7@. D9 7(*9=@ (; :7 ;9;(:7>*=7
9<:97@7 9; 9>7 :;97. ф a= ф ' m=arctan
•
ca
(
tanф' F
)
9 :7 (K9?; ?:?7@7
c a= c ' m=
c ' F
C <7=H9>= 7@?9;?;7:+ 9; 9: 99 @9 =@9;7@7 9 9<:97 9: ,&ero de es#a)ilidad0 ?;9= @9: (9?(?9;>9 @9 9>78?:?@7@ @9: H87( @9 T7:= 7;>9=?=. N = c ' m =
c ' F
C <*9@9 7<=9(?7=9+ 7 ?*7:@7@ @9 (?=(*;>7;(?7 (*7;> 7= 97 N+ 9;= =9*:>7 9: 7(>= @9 9*=?@7@. A*;*9 9: ? H87( ?;(:*9 *;7 :99;@7 9<:?(7>?7 '(7 1+ 2 3-+ * ?;>9=<=9>7(?; 9<:9 ; =9*:>7; ?;9@?7>+ @9 7;9=7 *9 =9*:>7 @9 ?;>9= =97:?7= *;7 @9(=?<(?; 7: H @9>7::7@7 '; 9 ?;(:*9; 9; 9>7 @9(=?<(?; :7 ?>*7(?;9 @9: H87( (; ф a= 0
+ *9 7 9 K7; ?;(:*?@ 9; :7 @9(=?<(?; @9: (7 ?; @=9;79-.
P7=7 9::+ 9; <=?9= :*7= 9 9*?=H; *(9?79;>9 :7 :;97 @9: H87( (==9<;@?9;>9 7 : @?>?;> =7?9;> ?:?7@ ф ' a . P>9=?=9;>9 9 (9;>7=7; : (7 9; : *9 <7=9(9 9?>?= *;7 @*<:?(?@7@ @9 <?8?:?@7@9 ':;97 @?9=9;>9 <7=7 : ? =7?9;> ?:?7@-.
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MODELO DE FALLA CIRCULAR
Figura 27. 6baco de 0aylor para suelos con coesión y rozamiento ("%?%)
1B8C% "E !%E= ; B)8; C K9 9;(?;7@ 9; 9: 77;(9 @9: <=99;>9 >=787+ (*7;@ 9: 7(? =( 9 9;(*9;>=7 *9=>99;>9 7:>9=7@ =7(>*=7@ <*9@9 ::97= 7 (<=>7=9 ( ? @9 *; 0*9: K;9 9 >=7>7=7. E; 9>7 (?=(*;>7;(?7 :7 >?<:7 @9 7::7 H <=878:9 9=H :7 circular. P7=7 9>*@?7= 9>9 >?< @9 =>*=7 9\ B=7 9:78=7=; : H87( (* (H:(*: <7=7 * 8>9;(?; 9 877; 9; 9: >@ @9: (=(*: @9 =7?9;>+ 9 @9(?=+ 7*?9;@ *9 :7
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MODELO DE FALLA CIRCULAR
>9;?;9 ;=7:9 9; :7 *<9=?(?9 @9 @9:?7?9;> 9 (;(9;>=7; 9; *; : <*;>+ <=<=(?;7;@ 7 *; :?>9 ?;9=?= @9: 7(>= @9 9*=?@7@. E> H87( ?;(=<=7; (?9=>7 (;?@9=7(?;9 @9 7*7 *8>9==H;97 @9 ?;@*@78:9 ?;>9=
a/ a e+istencia de presiones intersticiales en el seno del talud D98?@7 7 :7 <=99;(?7 @9 *; ;?9: =9H>?( 9; 9: >9==9;+ *9 @??@9 9: >7:*@ 9; *;7 ;7 9(7 >=7 7>*=7@7. 'F?*=7 2"b/ Posición de la l0nea de la l0nea de saturación L7 :;97 @9 7>*=7(?; (?;(?@9 (; :7 *<9=?(?9 *<9=?= @9: >7:*@+ *<;?9;@ 9; 9: >9==9; :7 9?>9;(?7 @9 *;7 =9@ @9 ?:>=7(?; 9; =?9; 9>7(?;7=?. 'F?*=7 2!A<7=>9 @9 :7 (;?@9=7(?;9 K?@=9:?(7 9;(?;7@7 9 K7; 7*?@ :7 ?*?9;>9 ?<:??(7(?;9 •
• • •
E: 7>9=?7: (;>?>*>? @9: >7:*@ 9 (;?@9=7 K;9 9; >@7 :7 9>9;?; @9: ?. E: (=(*: @9 =>*=7 9 K7(9 <77= ?9<=9 <= 9: 7:*@. L7 =9?>9;(?7 @9: >9==9; ?9;9 @7@7 <= 9: (=?>9=? @9 =>*=7 MK=C*:8. S9 (;?@9=7 :7 9?>9;(?7 @9 *;7 =?9>7 @9 >=7((?; *9 <*9@9 9>7= ?>*7@7 <= 9;(?7 <= @987 @9 :7 (=9>7 @9: >7:*@.
S9 >?9;9 9; (*9;>7 *9 9>7 =?9>7 @9 >=7((?; 9 ?>7 9; 9: 8=@9 *<9=?= @9: >7:*@ 7@9H 9; 9: :*7= H @97=78:9. E>7 =?9>7 @9 >=7((?; 9 *;7 @9 :7 (;@?(?;9 *9 K7(9; ;? 9: 7(>= @9 9*=?@7@ <7=7 (7@7 >7:*@ @9>9=?;7@ 7@9H @9: (9;>= @9: (9;>= @9: (=(*: @9 =>*=7.
Figura 28. Red de filtración en el talud cuando la línea de saturación coincide con la superficie del terreno. (;oe y Bray! "#$$)
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MODELO DE FALLA CIRCULAR
Figura 29. *istintas situaciones de la línea de saturación consideradas en los bacos. (;oe y Bray "#%") A @?9=9;(?7 @9 : H87( @9 T7:= *9 *<;9; *; >7:*@ 9( ?; ;?9: =9H>?(+ : H87( @9 9\ B=7 ; H >?:9. EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
42
MODELO DE FALLA CIRCULAR
imitaciones en los bacos de !oe5 2 Bra2: •
L7 7<=?7(?; @9 9> H87( ; 9 H:?@7 <7=7 H;*: @9 =7?9;> ?;>9=; ˂
•
5W. D9: <*;> 7;>9=?=+ 9 9;>?9;@9 *9 : H87( 9(:*9; :7 =>*=7 <=@*(?@7 9; (;@?(?;9 ?; @=9;79 (=> <:7 ' c = Su + фu=0 -+ <7=7 :7 *9 9 K7 ?> *9 : (=(*: H @97=78:9 <*9@9; ; 9= @9
E; :7 ?*=7 2!+ ?;@?(7 :7 7:>*=7 @9: >7:*@ / :7 @?>7;(?7 9;>=9 9: 7:*@ 9: <*;> @9 (=>9 @9: ;?9: =9H>?( (; :7 *<9=?(?9 @9: >9==9;. S9 (*9;>7 (; 5 H87( 'F?*=7 31 7 35-+ @ @9 9:: <7=7 9: >7:*@ >>7:9;>9 9( >>7:9;>9 7>*=7@ 3 <7=7 (7 ?;>9=9@? '@?9=9;>9 7:>*=7 @9: ;?9: =9H>?( ?> 9; :7 F?*=7 2"-. L <7=H9>= <=99;>9 9; : H87( ; : ?*?9;>9
!:
A:>*=7 @9: >7:*@.
c:
CK9?; 99(>?7 @9: >9==9;.
,:
$;*: @9 =7?9;> ?;>9=; 99(>?.
γ
F:
: P9 9<9(?( @9: >9==9;. F7(>= @9 9*=?@7@
El proceso a seguir para el empleo de los bacos> Figura?@/ es el siguiente: 1- C;(?@ 9: =?9; @9 <=9?; ?;>9=>?(?7: 9; 9: >7:*@+ 9 9:9((?;7 9; :7 ?*=7 2" :7 ?>*7(?; *9 H 9 7<=?7 7 :7 =97:?@7@+ : *9 <=<=(?;7 9: H87( 7 9<:97=.
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MODELO DE FALLA CIRCULAR
2- E; 9: H87( 9:9((?;7@ 9 @9>9=?;7 9: <7=H9>= 7@?9;?;7:
c ' γ . H . tan ϕ
9 :(7:?7 9: 7:= ;*=?( 8>9;?@ 9; 9: 8=@9 (?=(*:7= 9>9=?= @9: H87(. 3- A <7=>?= @9 9>9 <*;> 9 =9(==9 9: =7@? @9: H87( K7(?7 9: =?9; @9 (=@9;7@7+ K7>7 ?;>9=(9<>7= :7 :;97 *9 =9<=99;>7 9: H;*: @9 ?;(:?;7(?; @9: >7:*@. 4- D9@9 9: <*;> @9 ?;>9=9((?; 9 >=77 *;7 K=?;>7: *;7 9=>?(7:+ *9 <=<=(?;7 =9<9(>?79;>9 : <7=H9>= 7@?9;?;7:9 tan ϕ
F
c ' γ.H . F
C*7:*?9=7 @9 : @ ; ?=9 <7=7 @9<97= 8>9;9= 9: (9?(?9;>9 @9 9*=?@7@ F.
Figura ?@. P=(9@??9;> <7=7 @9>9=?;7= 9: 7(>= @9 9*=?@7@ @9 *; >7:*@ '9\ B=7+ 1"!1-
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MODELO DE FALLA CIRCULAR
Figura 31. 6baco NC" (;oe y Bray! "#%")
Figura 32. 6baco NCD (;oe y Bray! "#%") EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
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MODELO DE FALLA CIRCULAR
Figura 33. 6baco NCE (;oe y Bray! "#%")
Figura 34. 6baco NC? (;oe y Bray! "#%") EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
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MODELO DE FALLA CIRCULAR
Figura 35. 6baco NCF (;oe y Bray! "#%") 9\ 8=7 9; * :?8= 0R(\ S:<9 E;?;99=?; ?;(:*9 >78?; H87( <7=7 <?(?;7= 9: (9;>= @9: (=(*: @9 @9:?7?9;> =?9>7 @9 >=7((?;.
$aludes drenados sin ni6el fretico/:
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MODELO DE FALLA CIRCULAR
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MODELO DE FALLA CIRCULAR
Figura 36. Gocalización del centro del círculo y la 'rieta de tracción en un talud drenado (;oe y Bray! "#%")
$aludes con ni6el fretico:
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MODELO DE FALLA CIRCULAR
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MODELO DE FALLA CIRCULAR
Figura 37. Gocalización de la 'rieta de tracción y centro de círculo. &ara situación del nivel fretico del baco NCE (;oe y Bray! "#%")
&.2 M#TODO DE DOVELAS O REBANADAS Fundamento del m*todo. C 9 K7 ?> 7;>9=?=9;>9+ 9: >@ @9 (=(*: @9 =7?9;> @97==::7@ <7=7 9>*@?7= 9: 9*?:?8=? :87: @9 *;7 77 *9: <>9;(?7:9;>9 ?;9>78:9 9 9;(;>=787 7>9H>?(79;>9 ?;@9>9=?;7@ 7: 9?>?= *;7 7= ;9= @9 ?;(;?>7 *9 @9 9(*7(?;9. P7=7 9?>7= 9>9 99(> 9=7 ;9(97=? =97:?7= 7:*;7 K?<>9? 8=9 :7 @?>=?8*(?; @9 >9;?;9 99(>?7 ;=7:9 7 : :7= @9 :7 *<9=?(?9 @9 @9:?7?9;>. C; 9: ?; @9 =7(?;7:?7= 9>7 K?<>9?+ F9::9;?* <:7;>9 9>*@?7= 9: 9*?:?8=?+ ; @9 >@7 :7 77 <>9;(?7:9;>9 @9:?7;>9+ ?; @9 *;7 9=?9 @9 77 =987;7@7 9=>?(7:9 9; :7 *9 @?(K7 77 9 @??@?=7. 'F?*=7 3!-.
Figura 38. *ivisión en faHas de una masa de suelo potencialmente inestable.
EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
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MODELO DE FALLA CIRCULAR
L7 ?@97 <=?9;9 @9: =7;7?9;> ?;>*?>? @9 *9 :7 >9;?; ;=7: 9; *; <*;> (*7:*?9=7 @9 *;7 *<9=?(?9 @9 @9:?7?9;> K7 @9 @9<9;@9= *;@79;>7:9;>9 @9: <9 @9 :7 77 *9 =7?>9 8=9 :. D9 9>7 7;9=7+ @??@?9;@ :7 77 @9 <>9;(?7:9;>9 @9:?78:9 9; 77 9=>?(7:9 *?(?9;>99;>9 <9*9Q7 '9 @9(?= 9; *; ;*9= *?(?9;>99;>9 =7;@9 @9 77-+ 9 <*9@9 7*?= *9 :7 *9=7 ;=7:9 9; (7@7 77 7(>7; 9; 9: <*;> 9@? @9 * 879. E; 9: 9>*@? @9: 9*?:?8=? @9 :7 77 9 (;?@9=7=7 *9 :7 *9=7 ?;>9=7(>7; 9;>=9 9::7+ (7 *9 ; (*==9 9; : >@ @9 77 :87:. D9 K9(K+ *;7 @9 :7 @?9=9;(?7 9;>=9 : >@ @9 @9:7 '77- : @9 9*?:?8=? :87: 9 :7 ;9(9?@7@ @9 =97:?7= K?<>9? 8=9 :7 *9=7 @9 ?;>9=7((?; 9;>=9 77+ 9; :*7= @9 :7 @?>=?8*(?; @9 >9;?;9 ;=7:9 7 : :7= @9 :7 :;97 @9 @9:?7?9;>. C 9+ : >@ @9 77 ; >78?; @9 9*?:?8=? :?>9 <= : >7;> =9*?9=9; <>*:7= *;7 @9>9=?;7@7 *<9=?(?9 @9 @9:?7?9;>+ <7=7 :7 *9 9 (7:(*:7 9: 7(>= @9 9*=?@7@.
Aenta7as del m*todo de do6elas: •
L <7=H9>= @9 =9?>9;(?7 7: (=>9 '(+ ф- 7 : :7= @9 :7 *<9=?(?9 @9 @9:77?9;> '879 @9 77- 9 <*9@9; @??(7= @9 77 7 77+ @9 7;9=7 *9 9
•
<?8:9 (;?@9=7= >7:*@9 ; K;9 (; @?9= >?< @9 >9==9;. A:*; @9 : >@ ; =9*?9=9; *9 :7 *<9=?(?9 @9 @9:?7?9;> 7 >7;>97= 97; (?=(*:7=9+ @9 =7 *9 7(?:?>7; 9: 7;H:?? @9 =7 @9 7::7 89=;7@7 <= K9>9=9;9?@7@9 9:?(7 9>9=9=H?(7.
7@ *; >7:*@ (; *<9=?(?9 @9 @9:?7?9;> (?=(*:7=+ @9 :7 *9 9 K7 9>=7@ *;7 =987;7@7 '77-. S9 K7; =9<=99;>7@ 7?? :7 *9=7 7:*; 7<9(> 9>=?( 9; *;7 =987;7@7.
EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
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MODELO DE FALLA CIRCULAR
Figura 39.
Aariables geom*tricas: •
•
•
•
•
b!
7;(K @9 :7 77.
"!
:;?>*@ @9 :7 879 @9 :7 =987;7@7
# !
H;*: *9 =7 :7 ?;(:?;7(?; @9 :7 77 (; :7 K=?;>7:
$!
7:>*=7 9@?7 @9 :7 =987;7@7.
% !
8=7 @9: <9 @9 :7 =987;7@7 (; =9<9(> 7: (9;>= @9: (=(*:. *9 ?9;9 7
9= ?*7: 7 R . sin # ! •
&
H;*: (9;>=7: @9: (=(*: @9 @9:?7?9;>.
Fuerzas actuantes: •
W !
P9 @9 :7 =987;7@7 EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
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MODELO DE FALLA CIRCULAR
•
•
•
!
=9*:>7;>9 @9 :7 <=9?;9 ?;>9=>?(?7:9 *9 7(>7; 9; :7 879 @9 :7 =987;7@7.
N ' !
=9*:>7;>9 @9 :7 >9;?;9 99(>?7 ;=7:9 9; :7 879 @9 :7 =987;7@7.
Rm .!
=9*:>7;>9 @9 :7 >9;?;9 >7;9;(?7:9 ?:?7@7 9; :7 879 @9 :7
=987;7@7. D7@ *9 9: (=?>9=? @9 7::7 7@<>7@ 9 9: @9 MK=C*:8+ =9*:>7 T ! =
tan ф ' c ' "! + N ' ! F F
D;@9 ( ф ; :7 (K9?; =7?9;> ?;>9=; 99(>? 9; <:7; @9 :7 879 @9 :7 =987;7@7.
9ota: E; : >@ 7;>9 ?> Rm
9=7 =9<=99;>7@ (
T + 9 @9(?=
78 ; : ?+ : *9 7* 9; 9: >@ @9 @9:7 >=7877=9 (; :7 7(9<(?; @9 Rm . •
•
( ! ) ( !+1
=9*:>7;>9 @9 :7 >9;?;9 >7;9;(?7:9 9; :7 (7=7 9=>?(7:9 @9 *;
>= :7@ @9 :7 =987;7@7. *! ) * !+ =9*:>7;>9 @9 :7 >9;?;9 ;=7:9 9; :7 (7=7 9=>?(7:9 @9 *; >= 1
:7@ @9 :7 =987;7@7.
S*<;?9;@ 7K=7 *9 9: >7:*@ 9 @??@9 9; n =987;7@7+ 9 >9;@=7; :7 ?*?9;>9 ?;(;?>7
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MODELO DE FALLA CIRCULAR
C; =9:7(?; 7 :7 9(*7(?;9+ 9 @?<;9; :7 ?*?9;>9
C <*9@9 7<=9(?7=9+ 9?>9; '6; 2- 4; ) n H ?;(;?>7 *9 9(*7(?;9+ :*9 <7=7 =9:9= 9: <=8:97 9 ;9(97=? =97:?7= 2; 2 K?<>9? 7@?(?;7:9. L >@ @?<;?8:9 9 @?9=9;(?7 9; :7 K?<>9? *9 9 =97:?7;+ 9 @??@9 9; >@ 7<=?7@ >@ <=9(? (<:9>. 'F?*=7 4&-
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MODELO DE FALLA CIRCULAR
O
Figura 4$. M,todos de la rebanada (8lalla "###)
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MODELO DE FALLA CIRCULAR
&.2.1 M#TODOS APRO%IMADOS
S; 7*9:: 9; : *9+ <= 9: ;9= @9 K?<>9? =97:?7@7 ; 9 ::97; 7 7>?7(9= >@7 :7 9(*7(?;9 @9 9*?:?8=?. L H =9:97;>9 <= * @?*?; <=H(>?(7 ; : ?*?9;>9 M#TODO DE FELLENIUS O CONVENCIONAL
F9::9;?* '1"36- @97==:: 9: <=?9= >@ @9 (H:(*: @9 9>78?:?@7@ @9 >7:*@9 9@?7;>9 * @???; 9; =987;7@7 9=>?(7:9. L7 *<9=?(?9 @9 @9:?7?9;> 9 *<;9 (?=(*:7=+ :7 (;@?(?; *;@79;>7: <7=7 :7 8>9;(?; @9: (9?(?9;>9 @9 9*=?@7@ 9 877 9; 9: 9*?:?8=? @9 9;>. E: >@ <7=>9 @9 :7 K?<>9? 7 9;(?;7@7 @9 *9 9: <*;> @9 7<:?(7(?; N+ 9 ?>7 9; 9: (9;>= @9 879 @9 (7@7 =987;7@7+ : *9 @7 :*7= 7 *9 :7 =9*:>7;>9 N != N ' ! + ! + 9; (7@7 *;7 @9 9::7 @989 <77= <= 9: (9;>= @9: (?=(*: O. A@9H+ :7 *7 @9 : 9;> @9 :7 *9=7 9;>=9 =987;7@7 7 : :7= @9 >@ 9: >7:*@ =9*:>7 ;*:7+ (; : *9 9: <=8:97 *9@7 87>7;>9 ?<:??(7@. A+ >7;@ 9;> (; =9<9(> 7 O+ :7 ;?(7 *9=7 ?<:?(7@7 ; : <9 :7 =9*:>7;>9 @9 =9?>9;(?7 ?:?7@7
E: (9?(?9;>9 F 9 @9>9=?;7 7 <7=>?= @9: 9*?:?8=? @9 9;>. A*?9;@ 7K=7 *9 9: (9?(?9;>9 @9 9*=?@7@ 9 ;?( 7 : :7= @9 :7 *<9=?(?9 9 <*9@9 @9<97= >9 @9 :7 9<=9?; 7;>9=?=+ =9*:>7;@. E(*7(?; 1
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MODELO DE FALLA CIRCULAR
P7=7 @9>9=?;7= 7 (;>?;*7(?; :7 *9=7 ;=7: 9; :7 879 @9 :7 =987;7@7+ N ! + @9 : @ 9(*7(?;9 @9 9*?:?8=? @9 *9=7 @?<;?8:9. F9::9;?* 9:?? (7:(*:7= : 9: 9*?:?8=? 9; *;7 @?=9((?;+ :7 <9=<9;@?(*:7= 7 :7 879 @9 (7@7 =987;7@7. E(*7(?; 2 C; : *9 *>?>*9;@ 9>7 9(*7(?; 9; :7 7;>9=?=+ ; =9*:>7
A:7 ?>7 @9 9>7 (<:?(7@7 9(*7(?;+ 9; :7 *9 7: <=?;(?=9 =987;7@7 ( ! *! + F9::9;?* @9(?@? =97:?7= :7 K?<>9? ?<:??(7@=7 @9 *9
%*9 9; =97:?@7@ 9*?7:9 7 *<;9= *9 >@7 :7 *9=7 9;>=9 =987;7@7 ; ;*:7
E; (;9(*9;(?7+ 9: 7(>= @9 9*=?@7@ =9*:>7 E(*7(?; 3
E; =9*9;+ :7 K?<>9? 7@?(?;7:9 =97:?7@7 <= F9::9;?* *9=;
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MODELO DE FALLA CIRCULAR
L *9 *<;9 K789= =97:?7@ n K?<>9? H @9 :7 9>=?(>79;>9 ;9(97=?7 '*;7 <= =987;7@7-. E>9 >@ *9 ( <*9@9 7<=9(?7=9 =9*:>7 * 9;(?:: 9; * 7<:?(7(?; >7; : =9*?9=9 :7 =97:?7(?; @9 *; *7>=? 7 07; 9@?7;>9 *;7 K7 @9 (H:(*:+ =9*:>7 7(9<>78:9 ? :7 7=?7(?; 9: H;*: # ! 9 @?(=9>7+ : *9 9 : ?+ ? 9: H;*: (9;>=7: & 9 =9:7>?79;>9 <9*9Q. E; (7 (;>=7=? 9: 99(> @9 :7 <=9?; ?;>9=>?(?7: 9 7;??(7 <*9@9 @7= :*7= 7 7:=9 @9 N ' ! 9(9?79;>9 87+ ?;(:* ;97>?+ : *9 =9@*(9 9: 7(>= @9 9*=?@7@ 8>9;?@. M#TODO DE BISOP SIMPLIFICADO
E; 1"54 B?K< @97==:: *; <=(9@??9;> ??:7= 7: @9 F9::9;?*+ ?;>=@*(?9;@ *;7 7=?7;>9 @9 ?<=>7;(?7. L7 (;?@9=7(?;9 ?;?(?7:9 9; (*7;> 7 :7 <?(?; @9 :7 *9=7 N ' ! 7 :7 9:9((?; @9: 9*?:?8=? @9 9;> ( (;@?(?; @9 9*?:?8=? *;@79;>7: =9*:>7; ?*7:9 7 :7 <=<*9>7 @9 F9::9;?*+ (; : *9 9: @97==:: 9 ?@;>?( K7>7 :7 8>9;(?; @9 :7 9(*7(?; 1
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MODELO DE FALLA CIRCULAR
C; =9<9(> 7: 9*?:?8=? @9 *9=7+ B?K< 9:9((?; >78?; *;7 :7 @?=9((?;+ <9= 9; 9>9 (7 *9 :7 9=>?(7: <= 9: (9;>= @9 (7@7 =987;7@7. E; 9>7 (;@?(?;9 :7 9(*7(?; @9 9*?:?8=? (==9<;@?9;>9 =9*:>7 'F?*=7 3"-
Y *>?>*9;@ Rm .!
D9<97;@ N ' ! @9 9>7 9<=9?;
E(*7(?; 4
L:77;@ 7K=7
+ T != ( ! − ( !+ 1
! =9(=@7;@ *9
b!= "! . cos # !
*>?>*9;@ :7
9(*7(?; 4 9; :7 9(*7(?; 3+ >9;9
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MODELO DE FALLA CIRCULAR
L7 9<=9?; 7;>9=?= (==9<;@9 7: >@ 0H =?*= @9 B?K<. C <*9@9 7<=9(?7=9+ 9: 7(>= @9 9*=?@7@ 9 9;(*9;>=7 ?<:(?> 9; :7 9(*7(?;+ : *9 8:?7 7*?= *; 7(>= @9 9*=?@7@ ?;?(?7: ::97= 7 (78 7=?7 ?>9=7(?;9 K7>7 *9 :7 :*(?; (;9=9. P= >=7 <7=>9+ 9; :7 9(*7(?; 7;>9=?= ?*=7; :7 *9=7 9=>?(7:9 9;>=9 =987;7@7. D7@ *9 9; <=?;(?@7 ;*:7 ( ∀!, T ! =0 ) + : *9 @? :*7= 7: ::77@ 0>@ ?<:??(7@+ *9 >78?; K7 @9 =9:9=9 <= ?>9=7(?;9 (*7 9<=9?; =9*:>7
E(*7(?; 5
S9 @9?;9 *; <7=H9>= M # + *9 =9(9 ?<:(?>79;>9 9: 7:= @9: 7(>= @9 9*=?@7@+ @9 9>7 7;9=7 7*@H;@9 @9 :7 ?*=7 41 9 <*9@9 (;(9= 9: 7(>= @9 9*=?@7@ @9 *; =>*=7 (?=(*:7=+ (;(?9;@ 9: H;*: @9 =7?9;> @9 :7 *<9=?(?9 @9 =>*=7 9: H;*: =.
E;>;(9 :7 9(*7(?; 5 *9@7 @9 :7 ?*?9;>9 7;9=7
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MODELO DE FALLA CIRCULAR
M # del parmetro 41. 8btención P7=7 ?;>=@*(?= @9 ;*9 Figura :7 *9=7 9;>=9 =987;7@7+ B?K< <=<* 7*>7=:7 9@?7;>9
?>9=7(?;9 K7>7 (*<:?= :7 n 9(*7(?;9 @9 9*?:?8=? K=?;>7:. S?; 987= 9>9 <=(9 =9*:>7 @?(?:+ @9 =7 *9 9: <=(9@??9;> *9 =97:9;>9 9 @?*;@? @9 =7 *;?9=7: *9 9: 0?<:??(7@. R9*?9;@+ :7 K?<>9? 7@?(?;7:9 =97:?7@7 <= B?K< <7=7 9: (7 ?<:??(7@ 9=?7;
L *9 *<;9 K789= =97:?7@ 1 K?<>9? H @9 :7 9>=?(>79;>9 ;9(97=?7+ @7;@ :*7= 7 :7 ;9(9?@7@ @9 =9:9= <= ?>9=7(?;9. P7=7 ?;7:?7=+ 9 ?;>9=97;>9 K7(9= ;>7= *9 (; 9>9 >@ @9 8=99>?7(?; @9: 99(> @9 :7 <=9?;9 ?;>9=>?(?7:9 <7=7 H;*: (9;>=7:9 9:97@ ; 9 >7; 99= ( 9; 9: >@ @9 F9::9;?*+ <=<=(?;7;@ (9?(?9;>9 @9 9*=?@7@ H 9:97@ <=? 7 :7 =97:?@7@. L7 ?*=7 42 *9>=7 :7 (<7=7(?;9 =97:?7@7 <= B?K< '1"55- 9;>=9 78 <=(9@??9;> <7=7 @?9= 7:=9 @9: H;*: (9;>=7: @9: 7(>= @9 <=9?; ?;>9=>?(?7: = *.
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MODELO DE FALLA CIRCULAR
Figura 42. omparación entre los m,todos de 5ellenius y Bisop simplificado con relación al n'ulo central del círculo de deslizamiento y el factor ru de presión intersticial. (Bisop! "#FF) M#TODO SIMPLIFICADO DE JANBU
E; : >@ @9 F9::9;?* B?K< :7 (;@?(?; *;@79;>7: 7 7>?7(9= 9 :7 @9 9*?:?8=? @9 9;>. D9 K9(K+ 9; ;?;*; @9 : @ (7 9 7>?7(9 9: 9*?:?8=? @9 *9=7 K=?;>7:9. E; 9>9 >@ 9 7*9 *9 :7 *9=7 >7;9;(?7:9 9; :7 (7=7 @9 :7 77 ; ;*:7 ' ( != 0
-+ <9= 9; 9>9 (7 :7 9(*7(?; *9 ; 7>?7(9 (<:9>79;>9 9: 9*?:?8=? 9 :7
@9 9;>. S?; 987=+ J7;8* ?;>=@*(9 *; 7(>= @9 (==9((?; - + <7=7 (<9;7= 0
9>9 <=8:97+ ?> 9; @9>7::9 H 7@9:7;>9. E>9 >@ <=99;>7 :7 9;>77 8=9 : 7;>9=?=9 @9 *9 ; 9?9 *9 :7 *<9=?(?9 @9 =>*=7 97 (?=(*:7=. 'F?*=7 43-
Figura 43. Mecanismos de falla a) deslizamiento rotacional! b) rotura traslacional. EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
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MODELO DE FALLA CIRCULAR
E; : *9 =9<9(>7 7: <:7;>97?9;> @9: >@+ 7: ?*7: *9 9; 9: @9 B?K<+ 9 (;?@9=7 9: 9*?:?8=? @9 *9=7 9=>?(7:9 9; (7@7 =987;7@7. A?? 9 *<;9 *9 + T != ( ! − ( !+ 1=0
+ : *9 @7 :*7= 7: >@ 0?<:??(7@ @9 J7;8*.
R9:?9;@ 9: 9*?:?8=? @9 :7 *9=7 K=?;>7:9+ 9 8>?9;9 :7 ?*?9;>9 9<=9?; <7=7 9: 7(>= @9 9*=?@7@.
E(*7(?; 6
D;@9
P>9=?=9;>9+ J7;8* '1"3- ?;>=@* *; 7(>= 9<=?( @9 (==9((?; ' - - 8=9 9: 0
(9?(?9;>9 @9 9*=?@7@ 7;>9=?= (; 9: ?; @9 9=7= : =9*:>7@. D?(K 7(>=+ @9<9;@?9;@ @9: >?< @9 7>9=?7: @9 :7 99>=7 @9: @9:?7?9;>. 'F?*=7 44-
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MODELO DE FALLA CIRCULAR
Figura 44. 5actor de corrección - para el m,todo de +ambu. 0
C 9; 9: (7 @9 B?K< ?<:??(7@+ 9: ;9= @9 K?<>9? 7@?(?;7:9 9 2; 1+ *;7 H @9 :7 ;9(97=?7.
S9 @9(=?8?=H : <=(9@??9;> @9 9>9 >@ *7;@ >78:7+ 7 *9 9 @9 H(?: 7<:?(7(?; aso 1 S9 @??@9 *; H=97 *9 =9<=99;>7 9: :*9; *;?>7=? @9 :7 77+ (*7 9>78?:?@7@ 9 @997 (;(9=+ 9; :9;9 <7=(?7:9 @9;?;7@ ( 7 9;(?;7 0@9:7 =987;7@7. P7=7 ?<:??(7= : (H:(*: 9 7 9(9 (;9;?9;>9 7*?= *; ? 9<9= EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
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MODELO DE FALLA CIRCULAR
@9 @9:7+ @9;?;7@ b. E: H;*: 9 ?@9 >=77;@ *;7 :;97 @9 <9;@?9;>9 9@?7 9; 9: (9;>= @9 :7 879 @9 :7 =987;7@7. 'F?*=7 45%
Figura 45 aso 2 C; 9: <9 :7 7:>*=7 9@?7 @9 :7 @9:7+ 9 @9>9=?;7 I *9 9 ?*7: 7 W = γ . $ . b + ::77;@ p= γ . $ . aso 3 D9>9=?;7(?; @9 :7 <=9?; @9 7*7 U. <7=7 9:: 9 (7:(*:7 9: <=9@? @9 :7 <=9?; @9 :7 879 @9 (7@7 @9:7 ?*?9;@ 9: ?*?9;>9 <=(9@??9;>
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MODELO DE FALLA CIRCULAR
Figura 46. *eterminación de la presión de a'ua J. aso 4 S9 K7(9 *;7 >78*:7(?; (; @7> @9 9;>=7@7 W = tan # +
T78?; ( =( c + ( p + u ) tan # ) . b aso 5 P7=7 : (H:(*: 9 7*9 *; 7(>= @9 9*=?@7@+ (7? ?9<=9 (; 9: 7:= @9 1+ <7=7 *; <=?9= >7;>9 @9 (H:(*:. E; >78:7 <=9@9>9=?;7@7 9 8>?9;9 : 7:=9 @9 n} rsub {α} ¿
+ @9;?;7@7 *;(?; 9>=?(7+ @9 :7 ?*=7 4 4! >=7@7 7
(;>?;*7(?; 7@9H 9 <;9; 9; :7 >78:7 : 7:=9 @9 /] n# + <7=7 (7@7 @9:7.
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MODELO DE FALLA CIRCULAR
Figura 47. *eterminación de los valores de n# para los valores positivos de #
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MODELO DE FALLA CIRCULAR
Figura 48. *eterminación de los valores de n# para los valores ne'ativos de #
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MODELO DE FALLA CIRCULAR
aso 6 S9 @9>9=?;7=7 9: 7(>= @9 (==9((?; @9 :7 ?*=7 4" 7 9 @9>9=?;7 9: ;*9 7(>= @9 9*=?@7@. E: 7(>= @9 9*=?@7@ 7 @9>9=?;7= 9@?7;>9 c + ( p −u ) tan ф
¿ ¿ n# ¿ ¿ ∑¿ F =- ¿ 0
S?<:??(7;@ (; :7 *>?:?7(?; @9 :7 9(*7(?;9 ?>7 9; 9: <7 4+ >9;9
∑ n( F =- ∑ p . tan # #
0
aso 7 S9 K7(9; ?>9=7(?;9. S? 9: 7(>= @9 9*=?@7@ 9; 9: <7 6 ; (;(*9=@7 (; 9: 7(>= @9 9*=?@7@ 7*?@ 9;9: <7 5+ 9 7*9 *; ;*9 7:= @9 F+ <7=9(?@ 7: 8>9;?@ 9; 9: <7 6. E;>;(9 9 =99; : <7 5 6+ >7;>7 9(9 K7>7 *9 9: 7:= (7:(*:7@ 97 ??:7= 7: 7*?@. F?; @9 : (H:(*:.
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MODELO DE FALLA CIRCULAR
Figura 49. Valores del factor de corrección (5i'ura ??)
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MODELO DE FALLA CIRCULAR
Figura5$. 0abulación de datos para el m,todo de +anbu
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MODELO DE FALLA CIRCULAR
&.2.2 M#TODOS PRECISOS O COMPLETOS
S9 @9;?;7; 7 : *9 (*<:9; >@7 :7 9(*7(?;9 @9 9*?:?8=?+ : *9 :9 <9=?>9 (;?@9=7= (*7:*?9= =7 9; :7 *<9=?(?9 @9 =>*=7. E; ;*9>= (7 (;?@9=7 <7=7 ?<:??(7= *9 :7 *<9=?(?9 @9 @9:?7?9;> 9 (?=(*:7=. A@<>7;@ :7 <=?9=7 K?<>9? K78?>*7: @9 *9 :7 *9=7 ;=7:9 N ' ! 9 :(7:?7; 9; 9: (9;>= @9 :7 879 @9 (7@7 =987;7@7 9<:97;@ :7 ?7 ;>7(?; *9 <7=7 9: (7 ?<:??(7@ @9 B?K<+ :7 9(*7(?; @9 9*?:?8=? @9 9;> =9*:>7=7 @9 ;*9
7(9<>7;@ *9 9: 7(>= @9 9*=?@7@ F =9<9(> 7: 9*?:?8=? @9 9;> 9 (;>7;>9 7 : :7= @9 >@ 9: >7:*@.
E(*7(?;
R9:?9;@ 7K=7 9: 9*?:?8=? K=?;>7: 9; >( 9: >7:*@ '7: =97:?7= :7 *7>=?7 :7 *9=7 9;>=9 =987;7@7 9 @97<7=9(9;-.
S*>?>*9;@ :7 9<=9?; @9 Rm .! 7(9<>7;@ *9 9: (9?(?9;>9 @9 9*=?@7@ (; =9<9(> 7: 9*?:?8=? @9 *9=7 K=?;>7:9 F 9 (;>7;>9 7 : :7= @9 >@7 :7 *<9=?(?9 @9 @9:?7?9;>.
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MODELO DE FALLA CIRCULAR
Y <= : >7;>
E(*7(?; !
L7 *9=7 ;=7: 9; :7 879 @9 :7 =987;7@7 9 @9>9=?;7+ ( 9; 9: (7 @9 B?K<+ 9@?7;>9 9: 9*?:?8=? 9; :7 9=>?(7:
E(*7(?; "
D;@9 F 9; :7 9(*7(?; " 9 F F + @9<9;@?9;@ @9 ? 9 (;?@9=7 9: 9*?:?8=? @9 9;> '9(*7(?; - @9 *9=7 '9(*7(?; !-. P7=7 =9:9= 9: <=8:97 9 ;9(9?>7 =97:?7= *;7 9=?9 @9 K?<>9? 7@?(?;7: (; =9<9(> 7 :7 *9=7 9;>=9 =987;7@7+ ?9;@ <=9(?79;>9 9>7 K?<>9? :7 *9 @?9=9;(?7 : @?9= &%#odos co&ple#os. M#TODO DE MORGENSTERN 6 PRICE '1&5
E; 9>9 <=?9= (7 9 *<;9 *9 :7 =9:7(?; 9;>=9 :7 *9=7 9;>=9 :7 =987;7@7 <*9@9 9<=97=9 9@?7;>9 :7 *;(?;.
D;@9 '/- 'F?*=7 51- @9(=?89 @9 7:*;7 7;9=7 :7 =7 @9 ( ! / *! + 7=?7 7 : :7= @9: >7:*@+ 9: (9?(?9;>9
'
0 < < 1
- 9 *; 7(>= @9 (==9((?; 7 @9>9=?;7=
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MODELO DE FALLA CIRCULAR
'?;(;?>7- <7=7 *9 9 (*<:7; :7 (;@?(?;9 @9 9*?:?8=? K=?;>7: @9 9;> 'F ) F -
Figura 51. 5unción f(K) C; 9>7 <=9?7+ :7 K?<>9? 7@?(?;7:9 =97:?7@7 ;
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MODELO DE FALLA CIRCULAR
C <*9@9 7<=9(?7=9+ 9 ?;>=@*(9 H @9 :7 9>=?(>79;>9 ;9(97=?7 '2; 2-. S?; 987=+
(;>?>*9 *;7 ?;(;?>7 7@?(?;7: 7 (7:(*:7= <7=7 *9 9 (*<:7 9:
9*?:?8=? @9 *9=7 9;>+ @9 7;9=7 *9 9: ?>97 *9@7 @9>9=?;7@ >?9;9 :*(?; 7>9H>?(7. M=9;>9=; P=?(9 '1"65- 9Q7:7;+ 9; * 9<=79 @9 (;(:*?;9+ *9 :7 *;(?; '/ <*9@9 9:9((?;7=9 7 <7=>?= @9: (;(??9;> 7<=?7@ @9 :7 @?>=?8*(?; @9 :7 >9;?;9 ?;>9=;7 9; 9: >7:*@+ @9 * ;?>=?7(?; '9 =9?9=9 7: (7 @9 <=97-. E; :7 <=H(>?(7 K78?>*7: *9:9; <=99:9((?;7=9 ?; 987= 7:*;7 =7 >7 :7 9(*7(?;9 (==9<;@?9;>9 8>9;?@7 *;7 :*(?;+ 9 <*9@9 =97:?7= :7 (<=87(?; @9 *9 : =9*:>7@ ; :?(+ 9 @9(?=+ *9 <= 99<:
L7 *9=7 ( ! ; 9(9@9; :7 H?7 ?:?78:9 9; 9: (=?>9=? @9 MK= C*:8
L <*;> @9 7<:?(7(?; @9 :7 *9=7 *! ; :?( '*9 7: 9; 9>; ?>*7@ @9;>= @9 :7 (7=7 @9 :7 =987;7@7-. L7 @?>=?8*(?; @9 *9=7 N ' ! 9 =7;78:9 7 : :7= @9 :7 :;97 @9 =>*=7 '; 9 <=@*(9; 0*;7@79;>9+ 7 9; * <=?9=7 <=<*9>7 @9 1"65 M=9;>9=; P=?(9 9Q7:787; *9 : 7(>=9 @9 9*=?@7@ 8>9;?@ ; *9:9; 9=9 9(9?79;>9 79(>7@ <= 9: >?< @9 *;(?; 9:9?@7+ : *9 :?(79;>9 =9>7 =9:97;(?7 7 * 9:9((?;+ 9 ?;(:* 7: =7;78:9 @9 :7 >9;?;9 ?;>9=;7 =9*:>7;>9 '9; >=?; @9 *! ( ! -.
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MODELO DE FALLA CIRCULAR
E: ;?( <=8:97 *9 <*9@9 <:7;>97=9 9; 7:*;7 (7?;9 9+ 9; >@ (7+ 9: @9 :7 (;9=9;(?7 ;*=?(7 ? @?(K7 >9;?;9 ; ; (<7>?8:9 (; 9: 9*?:?8=?+ 7;>9 : (*7: 9 ;9(97=? @??(7= :7 *;(?; '/- 9:9((?;7@7.
Figura 52. 5unciones f(K) abituales M#TODO DE SPENCER '1&!
E; 9>9 >@ 9 *<;9 *9 :7 =9*:>7;>9 @9 :7 *9=7 9;>=9 =987;7@7 >?9;9; *;7 ?;(:?;7(?; (;>7;>9 7 : :7= @9 >@ 9: >7:*@
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MODELO DE FALLA CIRCULAR
E> ?;??(7 *9 '/- ) 1 =tan / + <= >7;> 9 >=7>7 @9 *; (7 <7=>?(*:7= @9: >@ 7;>9=?=.
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MODELO DE FALLA CIRCULAR
Figura 53. M,todos de clculo de estabilidad de taludes 1ue emplean 5aHas Verticales
EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
79
MODELO DE FALLA CIRCULAR
1-.
CONCLUSIONES
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EAPIM UNMSM | LIZET GIOVANNA ORE SANTI
80
MODELO DE FALLA CIRCULAR
11.
BIBLIOGRAFÍA
E@*7=@ A:; P=9 @9 $=9@7 0E>78?:?@7@ @9 T7:*@9+ 2&&5. L*? O=>*Q A87@ UPM 0E>78?:?@7@ @9 >7:*@9 9; *9:. 2&&3. M7;*7: @9 >7:*@9 IGME. P9@= R7=9 O7;*=9; L97;@= A:97; M;9 0M9(H;?(7 @9 R(7 F*;@79;> 9 I;9;?9=7 @9 T7:*@9. F9=;7;@ 9==9=7 R@=*9 0A;H:?? @9 E>78?:?@7@ @9 T7:*@9 UCM+ 2&&&. M?*9: A;9: CKH9 M;(7 E>78?:?7(?; @9 T7:*@9 P=9(> CYTED /III+ 2&&5. R(\ S:<9 E;?;99=?; 9\ Y B=7. T9? D(>=7: S9>:7;7 M9:9;>?9?( 0E>78?:?@7@ @9 T7:*@9 9; M7(? R( (; (=?>9=? @9 =>*=7 ; :?;97:9 :99 @9 :*9;(?7 ; 7(?7@7+ 2&&5
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